From c531792ff499d19273b604a71527930ee19fe52b Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Thu, 19 Apr 2018 19:32:51 +0300 Subject: [PATCH] DM pour les 302 --- 3e/DM/DM_Paques/01_DM_18_05_03_302.tex | 278 +++++++++++++++++++++++ 3e/DM/DM_Paques/02_DM_18_05_03_302.tex | 278 +++++++++++++++++++++++ 3e/DM/DM_Paques/03_DM_18_05_03_302.tex | 278 +++++++++++++++++++++++ 3e/DM/DM_Paques/04_DM_18_05_03_302.tex | 278 +++++++++++++++++++++++ 3e/DM/DM_Paques/05_DM_18_05_03_302.tex | 278 +++++++++++++++++++++++ 3e/DM/DM_Paques/06_DM_18_05_03_302.tex | 278 +++++++++++++++++++++++ 3e/DM/DM_Paques/07_DM_18_05_03_302.tex | 278 +++++++++++++++++++++++ 3e/DM/DM_Paques/08_DM_18_05_03_302.tex | 278 +++++++++++++++++++++++ 3e/DM/DM_Paques/09_DM_18_05_03_302.tex | 278 +++++++++++++++++++++++ 3e/DM/DM_Paques/10_DM_18_05_03_302.tex | 278 +++++++++++++++++++++++ 3e/DM/DM_Paques/11_DM_18_05_03_302.tex | 278 +++++++++++++++++++++++ 3e/DM/DM_Paques/12_DM_18_05_03_302.tex | 278 +++++++++++++++++++++++ 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+\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 200 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 619 \euro pour le groupe puis 149 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 2000 & 5000 \\ + \hline + Prix au tarif Groupe en \euro & 619 & 2109 & 4344\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 200x$ + \item Tarif Groupe: $g: x \mapsto 149x + 619$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.5] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=5200, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{200*\x} + \tkzFct[domain=0:26, color=red, very thick]{149*\x+619} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $12.137254901960784$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 107m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 14m$ + \item $CD = 48m$ + \item $AE' = 19.1m$ + \item $AE = 38.2m$ + \item $AF = 15.3m$ + \item $E'F' = 25.95m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 14^2 + 48^2 \\ + AD^2 &= 196 + 2304 \\ + AD^2 &= 2500 \\ + AD &= \sqrt{2500} = 50m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 50 + 14 + 48 = 112m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 38.2 & AF = 15.3 & EF \\ + \hline + Triangle AE'F' & AE' = 19.1 & AF' & E'F' = 25.95 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{25.95 \times 38.2}{19.1} = 51.9 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 15.3 + 38.2 + 51.9 = 105.4m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 107m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 9 boules bleu, 2 boules jaunes, 2 boules vertes et 6 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{9}{19} \approx 0.47$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{11}{19} \approx 0.58$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{2}{19} \approx 0.11$ + + Boules rouges: $\dfrac{6}{19} \approx 0.32$ + + + Une boule rouge + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + B \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}J \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 8 & 4 & 1 & 1 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{1}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/02_DM_18_05_03_302.tex b/3e/DM/DM_Paques/02_DM_18_05_03_302.tex new file mode 100644 index 0000000..99e6a84 --- /dev/null +++ b/3e/DM/DM_Paques/02_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{02} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 112 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 435 \euro pour le groupe puis 83 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1120 & 2800 \\ + \hline + Prix au tarif Groupe en \euro & 435 & 1265 & 2510\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 112x$ + \item Tarif Groupe: $g: x \mapsto 83x + 435$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.8] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=2912, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{112*\x} + \tkzFct[domain=0:26, color=red, very thick]{83*\x+435} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $15.0$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 296m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 100m$ + \item $CD = 75m$ + \item $AE' = 8.7m$ + \item $AE = 26.1m$ + \item $AF = 33.6m$ + \item $E'F' = 78.23m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 100^2 + 75^2 \\ + AD^2 &= 10000 + 5625 \\ + AD^2 &= 15625 \\ + AD &= \sqrt{15625} = 125m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 125 + 100 + 75 = 300m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 26.1 & AF = 33.6 & EF \\ + \hline + Triangle AE'F' & AE' = 8.7 & AF' & E'F' = 78.23 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{78.23 \times 26.1}{8.7} = 234.7 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 33.6 + 26.1 + 234.7 = 294.4m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 296m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 2 boules bleu, 6 boules jaunes, 8 boules vertes et 6 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{2}{22} \approx 0.09$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{8}{22} \approx 0.36$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{8}{22} \approx 0.36$ + + Boules rouges: $\dfrac{6}{22} \approx 0.27$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + J \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}V \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & 4 & 6 & 4 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{6}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/03_DM_18_05_03_302.tex b/3e/DM/DM_Paques/03_DM_18_05_03_302.tex new file mode 100644 index 0000000..fd3d532 --- /dev/null +++ b/3e/DM/DM_Paques/03_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{03} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 151 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 453 \euro pour le groupe puis 109 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1510 & 3775 \\ + \hline + Prix au tarif Groupe en \euro & 453 & 1543 & 3178\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 151x$ + \item Tarif Groupe: $g: x \mapsto 109x + 453$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.6] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3926, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{151*\x} + \tkzFct[domain=0:26, color=red, very thick]{109*\x+453} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $10.785714285714286$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 60km. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 10km$ + \item $CD = 24km$ + \item $AE' = 12.35km$ + \item $AE = 24.7km$ + \item $AF = 12.2km$ + \item $E'F' = 8.55km$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 10^2 + 24^2 \\ + AD^2 &= 100 + 576 \\ + AD^2 &= 676 \\ + AD &= \sqrt{676} = 26km + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 26 + 10 + 24 = 60km + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 24.7 & AF = 12.2 & EF \\ + \hline + Triangle AE'F' & AE' = 12.35 & AF' & E'F' = 8.55 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{8.55 \times 24.7}{12.35} = 17.1 km + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 12.2 + 24.7 + 17.1 = 54.0km + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $ACDA$ car sa longueur est plus proche de 60km. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 5 boules bleu, 5 boules jaunes, 5 boules vertes et 10 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{5}{25} \approx 0.2$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{10}{25} \approx 0.4$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{5}{25} \approx 0.2$ + + Boules rouges: $\dfrac{10}{25} \approx 0.4$ + + + Une boule rouge + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + R \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}B \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 5 & 3 & 2 & 4 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{2}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/04_DM_18_05_03_302.tex b/3e/DM/DM_Paques/04_DM_18_05_03_302.tex new file mode 100644 index 0000000..f6d2951 --- /dev/null +++ b/3e/DM/DM_Paques/04_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{04} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 171 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 594 \euro pour le groupe puis 127 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1710 & 4275 \\ + \hline + Prix au tarif Groupe en \euro & 594 & 1864 & 3769\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 171x$ + \item Tarif Groupe: $g: x \mapsto 127x + 594$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.5] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=4446, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{171*\x} + \tkzFct[domain=0:26, color=red, very thick]{127*\x+594} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $13.5$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 297m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 100m$ + \item $CD = 75m$ + \item $AE' = 1.45m$ + \item $AE = 5.8m$ + \item $AF = 91.2m$ + \item $E'F' = 49.5m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 100^2 + 75^2 \\ + AD^2 &= 10000 + 5625 \\ + AD^2 &= 15625 \\ + AD &= \sqrt{15625} = 125m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 125 + 100 + 75 = 300m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 5.8 & AF = 91.2 & EF \\ + \hline + Triangle AE'F' & AE' = 1.45 & AF' & E'F' = 49.5 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{49.5 \times 5.8}{1.45} = 198.0 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 91.2 + 5.8 + 198.0 = 295.0m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 297m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 5 boules bleu, 2 boules jaunes, 7 boules vertes et 4 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{5}{18} \approx 0.28$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{7}{18} \approx 0.39$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{7}{18} \approx 0.39$ + + Boules rouges: $\dfrac{4}{18} \approx 0.22$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + R \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}B \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}R \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 3 & & 5 & 6 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{5}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/05_DM_18_05_03_302.tex b/3e/DM/DM_Paques/05_DM_18_05_03_302.tex new file mode 100644 index 0000000..972932f --- /dev/null +++ b/3e/DM/DM_Paques/05_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{05} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 151 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 498 \euro pour le groupe puis 115 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1510 & 3775 \\ + \hline + Prix au tarif Groupe en \euro & 498 & 1648 & 3373\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 151x$ + \item Tarif Groupe: $g: x \mapsto 115x + 498$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.6] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3926, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{151*\x} + \tkzFct[domain=0:26, color=red, very thick]{115*\x+498} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $13.833333333333334$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 300m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 144m$ + \item $CD = 17m$ + \item $AE' = 20.64m$ + \item $AE = 103.2m$ + \item $AF = 139.2m$ + \item $E'F' = 11.68m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 144^2 + 17^2 \\ + AD^2 &= 20736 + 289 \\ + AD^2 &= 21025 \\ + AD &= \sqrt{21025} = 145m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 145 + 144 + 17 = 306m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 103.2 & AF = 139.2 & EF \\ + \hline + Triangle AE'F' & AE' = 20.64 & AF' & E'F' = 11.68 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{11.68 \times 103.2}{20.64} = 58.4 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 139.2 + 103.2 + 58.4 = 300.8m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 300m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 9 boules bleu, 4 boules jaunes, 8 boules vertes et 3 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{9}{24} \approx 0.38$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{13}{24} \approx 0.54$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{8}{24} \approx 0.33$ + + Boules rouges: $\dfrac{3}{24} \approx 0.12$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + B \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}B \hspace{0.4cm}V \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 5 & 3 & 5 & 1 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{5}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/06_DM_18_05_03_302.tex b/3e/DM/DM_Paques/06_DM_18_05_03_302.tex new file mode 100644 index 0000000..65fcd80 --- /dev/null +++ b/3e/DM/DM_Paques/06_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{06} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 145 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 569 \euro pour le groupe puis 106 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1450 & 3625 \\ + \hline + Prix au tarif Groupe en \euro & 569 & 1629 & 3219\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 145x$ + \item Tarif Groupe: $g: x \mapsto 106x + 569$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.6] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3770, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{145*\x} + \tkzFct[domain=0:26, color=red, very thick]{106*\x+569} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $14.58974358974359$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 167m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 70m$ + \item $CD = 24m$ + \item $AE' = 0.57m$ + \item $AE = 1.7m$ + \item $AF = 36.7m$ + \item $E'F' = 41.4m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 70^2 + 24^2 \\ + AD^2 &= 4900 + 576 \\ + AD^2 &= 5476 \\ + AD &= \sqrt{5476} = 74m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 74 + 70 + 24 = 168m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 1.7 & AF = 36.7 & EF \\ + \hline + Triangle AE'F' & AE' = 0.57 & AF' & E'F' = 41.4 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{41.4 \times 1.7}{0.57} = 124.2 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 36.7 + 1.7 + 124.2 = 162.6m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $ACDA$ car sa longueur est plus proche de 167m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 10 boules bleu, 6 boules jaunes, 9 boules vertes et 7 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{10}{32} \approx 0.31$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{16}{32} \approx 0.5$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{9}{32} \approx 0.28$ + + Boules rouges: $\dfrac{7}{32} \approx 0.22$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + R \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}B \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 4 & 1 & 6 & 3 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{6}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/07_DM_18_05_03_302.tex b/3e/DM/DM_Paques/07_DM_18_05_03_302.tex new file mode 100644 index 0000000..17b8c8f --- /dev/null +++ b/3e/DM/DM_Paques/07_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{07} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 154 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 556 \euro pour le groupe puis 120 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1540 & 3850 \\ + \hline + Prix au tarif Groupe en \euro & 556 & 1756 & 3556\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 154x$ + \item Tarif Groupe: $g: x \mapsto 120x + 556$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.6] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=4004, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{154*\x} + \tkzFct[domain=0:26, color=red, very thick]{120*\x+556} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $16.352941176470587$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 118m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 28m$ + \item $CD = 45m$ + \item $AE' = 25.6m$ + \item $AE = 51.2m$ + \item $AF = 3.4m$ + \item $E'F' = 31.7m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 28^2 + 45^2 \\ + AD^2 &= 784 + 2025 \\ + AD^2 &= 2809 \\ + AD &= \sqrt{2809} = 53m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 53 + 28 + 45 = 126m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 51.2 & AF = 3.4 & EF \\ + \hline + Triangle AE'F' & AE' = 25.6 & AF' & E'F' = 31.7 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{31.7 \times 51.2}{25.6} = 63.4 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 3.4 + 51.2 + 63.4 = 118.0m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 118m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 8 boules bleu, 8 boules jaunes, 9 boules vertes et 8 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{8}{33} \approx 0.24$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{16}{33} \approx 0.48$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{9}{33} \approx 0.27$ + + Boules rouges: $\dfrac{8}{33} \approx 0.24$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + B \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}B \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}J \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 3 & 4 & 5 & 2 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{5}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/08_DM_18_05_03_302.tex b/3e/DM/DM_Paques/08_DM_18_05_03_302.tex new file mode 100644 index 0000000..f0d2117 --- /dev/null +++ b/3e/DM/DM_Paques/08_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{08} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 125 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 384 \euro pour le groupe puis 87 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1250 & 3125 \\ + \hline + Prix au tarif Groupe en \euro & 384 & 1254 & 2559\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 125x$ + \item Tarif Groupe: $g: x \mapsto 87x + 384$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.7] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3250, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{125*\x} + \tkzFct[domain=0:26, color=red, very thick]{87*\x+384} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $10.105263157894736$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 219m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 96m$ + \item $CD = 28m$ + \item $AE' = 36.15m$ + \item $AE = 72.3m$ + \item $AF = 7.7m$ + \item $E'F' = 69.8m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 96^2 + 28^2 \\ + AD^2 &= 9216 + 784 \\ + AD^2 &= 10000 \\ + AD &= \sqrt{10000} = 100m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 100 + 96 + 28 = 224m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 72.3 & AF = 7.7 & EF \\ + \hline + Triangle AE'F' & AE' = 36.15 & AF' & E'F' = 69.8 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{69.8 \times 72.3}{36.15} = 139.6 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 7.7 + 72.3 + 139.6 = 219.6m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 219m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 10 boules bleu, 7 boules jaunes, 10 boules vertes et 6 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{10}{33} \approx 0.3$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{17}{33} \approx 0.52$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{10}{33} \approx 0.3$ + + Boules rouges: $\dfrac{6}{33} \approx 0.18$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + V \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}J \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 6 & 3 & 5 & \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{5}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/09_DM_18_05_03_302.tex b/3e/DM/DM_Paques/09_DM_18_05_03_302.tex new file mode 100644 index 0000000..0163548 --- /dev/null +++ b/3e/DM/DM_Paques/09_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{09} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 184 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 688 \euro pour le groupe puis 146 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1840 & 4600 \\ + \hline + Prix au tarif Groupe en \euro & 688 & 2148 & 4338\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 184x$ + \item Tarif Groupe: $g: x \mapsto 146x + 688$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.5] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=4784, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{184*\x} + \tkzFct[domain=0:26, color=red, very thick]{146*\x+688} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $18.105263157894736$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 90km. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 24km$ + \item $CD = 32km$ + \item $AE' = 2.82km$ + \item $AE = 14.1km$ + \item $AF = 18.4km$ + \item $E'F' = 11.58km$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 24^2 + 32^2 \\ + AD^2 &= 576 + 1024 \\ + AD^2 &= 1600 \\ + AD &= \sqrt{1600} = 40km + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 40 + 24 + 32 = 96km + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 14.1 & AF = 18.4 & EF \\ + \hline + Triangle AE'F' & AE' = 2.82 & AF' & E'F' = 11.58 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{11.58 \times 14.1}{2.82} = 57.9 km + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 18.4 + 14.1 + 57.9 = 90.4km + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 90km. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 7 boules bleu, 6 boules jaunes, 4 boules vertes et 2 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{7}{19} \approx 0.37$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{13}{19} \approx 0.68$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{4}{19} \approx 0.21$ + + Boules rouges: $\dfrac{2}{19} \approx 0.11$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + J \hspace{0.4cm}B \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}V \hspace{0.4cm}B \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 6 & 4 & 4 & \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{4}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/10_DM_18_05_03_302.tex b/3e/DM/DM_Paques/10_DM_18_05_03_302.tex new file mode 100644 index 0000000..db7e122 --- /dev/null +++ b/3e/DM/DM_Paques/10_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{10} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 125 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 432 \euro pour le groupe puis 87 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1250 & 3125 \\ + \hline + Prix au tarif Groupe en \euro & 432 & 1302 & 2607\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 125x$ + \item Tarif Groupe: $g: x \mapsto 87x + 432$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.7] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3250, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{125*\x} + \tkzFct[domain=0:26, color=red, very thick]{87*\x+432} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $11.368421052631579$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 233m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 40m$ + \item $CD = 96m$ + \item $AE' = 13.47m$ + \item $AE = 40.4m$ + \item $AF = 113.6m$ + \item $E'F' = 26.47m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 40^2 + 96^2 \\ + AD^2 &= 1600 + 9216 \\ + AD^2 &= 10816 \\ + AD &= \sqrt{10816} = 104m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 104 + 40 + 96 = 240m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 40.4 & AF = 113.6 & EF \\ + \hline + Triangle AE'F' & AE' = 13.47 & AF' & E'F' = 26.47 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{26.47 \times 40.4}{13.47} = 79.4 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 113.6 + 40.4 + 79.4 = 233.4m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 233m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 8 boules bleu, 7 boules jaunes, 7 boules vertes et 4 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{8}{26} \approx 0.31$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{15}{26} \approx 0.58$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{7}{26} \approx 0.27$ + + Boules rouges: $\dfrac{4}{26} \approx 0.15$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + R \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}B \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}B \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 4 & 2 & 2 & 6 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{2}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/11_DM_18_05_03_302.tex b/3e/DM/DM_Paques/11_DM_18_05_03_302.tex new file mode 100644 index 0000000..1cf8338 --- /dev/null +++ b/3e/DM/DM_Paques/11_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{11} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 154 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 527 \euro pour le groupe puis 120 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1540 & 3850 \\ + \hline + Prix au tarif Groupe en \euro & 527 & 1727 & 3527\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 154x$ + \item Tarif Groupe: $g: x \mapsto 120x + 527$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.6] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=4004, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{154*\x} + \tkzFct[domain=0:26, color=red, very thick]{120*\x+527} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $15.5$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 339m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 140m$ + \item $CD = 51m$ + \item $AE' = 1.44m$ + \item $AE = 7.2m$ + \item $AF = 140.3m$ + \item $E'F' = 36.98m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 140^2 + 51^2 \\ + AD^2 &= 19600 + 2601 \\ + AD^2 &= 22201 \\ + AD &= \sqrt{22201} = 149m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 149 + 140 + 51 = 340m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 7.2 & AF = 140.3 & EF \\ + \hline + Triangle AE'F' & AE' = 1.44 & AF' & E'F' = 36.98 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{36.98 \times 7.2}{1.44} = 184.9 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 140.3 + 7.2 + 184.9 = 332.4m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $ACDA$ car sa longueur est plus proche de 339m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 10 boules bleu, 8 boules jaunes, 9 boules vertes et 5 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{10}{32} \approx 0.31$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{18}{32} \approx 0.56$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{9}{32} \approx 0.28$ + + Boules rouges: $\dfrac{5}{32} \approx 0.16$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + V \hspace{0.4cm}V \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}V \hspace{0.4cm}B \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}B \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 6 & 3 & 4 & 1 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{4}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/12_DM_18_05_03_302.tex b/3e/DM/DM_Paques/12_DM_18_05_03_302.tex new file mode 100644 index 0000000..51792bb --- /dev/null +++ b/3e/DM/DM_Paques/12_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{12} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 165 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 590 \euro pour le groupe puis 119 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1650 & 4125 \\ + \hline + Prix au tarif Groupe en \euro & 590 & 1780 & 3565\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 165x$ + \item Tarif Groupe: $g: x \mapsto 119x + 590$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.6] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=4290, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{165*\x} + \tkzFct[domain=0:26, color=red, very thick]{119*\x+590} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $12.826086956521738$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 175m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 48m$ + \item $CD = 55m$ + \item $AE' = 14.3m$ + \item $AE = 71.5m$ + \item $AF = 1.3m$ + \item $E'F' = 19.76m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 48^2 + 55^2 \\ + AD^2 &= 2304 + 3025 \\ + AD^2 &= 5329 \\ + AD &= \sqrt{5329} = 73m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 73 + 48 + 55 = 176m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 71.5 & AF = 1.3 & EF \\ + \hline + Triangle AE'F' & AE' = 14.3 & AF' & E'F' = 19.76 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{19.76 \times 71.5}{14.3} = 98.8 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 1.3 + 71.5 + 98.8 = 171.6m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $ACDA$ car sa longueur est plus proche de 175m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 8 boules bleu, 7 boules jaunes, 4 boules vertes et 9 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{8}{28} \approx 0.29$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{15}{28} \approx 0.54$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{4}{28} \approx 0.14$ + + Boules rouges: $\dfrac{9}{28} \approx 0.32$ + + + Une boule rouge + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + B \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}R \hspace{0.4cm}B \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}R \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 4 & 3 & 2 & 5 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{2}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/13_DM_18_05_03_302.tex b/3e/DM/DM_Paques/13_DM_18_05_03_302.tex new file mode 100644 index 0000000..245879f --- /dev/null +++ b/3e/DM/DM_Paques/13_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{13} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 163 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 604 \euro pour le groupe puis 124 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1630 & 4075 \\ + \hline + Prix au tarif Groupe en \euro & 604 & 1844 & 3704\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 163x$ + \item Tarif Groupe: $g: x \mapsto 124x + 604$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.6] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=4238, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{163*\x} + \tkzFct[domain=0:26, color=red, very thick]{124*\x+604} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $15.487179487179487$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 6km. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 4km$ + \item $CD = 3km$ + \item $AE' = 1.02km$ + \item $AE = 4.1km$ + \item $AF = 0.2km$ + \item $E'F' = 0.38km$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 4^2 + 3^2 \\ + AD^2 &= 16 + 9 \\ + AD^2 &= 25 \\ + AD &= \sqrt{25} = 5km + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 5 + 4 + 3 = 12km + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 4.1 & AF = 0.2 & EF \\ + \hline + Triangle AE'F' & AE' = 1.02 & AF' & E'F' = 0.38 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{0.38 \times 4.1}{1.02} = 1.5 km + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 0.2 + 4.1 + 1.5 = 5.8km + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 6km. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 3 boules bleu, 7 boules jaunes, 8 boules vertes et 8 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{3}{26} \approx 0.12$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{10}{26} \approx 0.38$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{8}{26} \approx 0.31$ + + Boules rouges: $\dfrac{8}{26} \approx 0.31$ + + + Une boule rouge + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + J \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}B \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}J \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 1 & 4 & 4 & 5 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{4}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/14_DM_18_05_03_302.tex b/3e/DM/DM_Paques/14_DM_18_05_03_302.tex new file mode 100644 index 0000000..a41baca --- /dev/null +++ b/3e/DM/DM_Paques/14_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{14} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 119 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 365 \euro pour le groupe puis 93 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1190 & 2975 \\ + \hline + Prix au tarif Groupe en \euro & 365 & 1295 & 2690\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 119x$ + \item Tarif Groupe: $g: x \mapsto 93x + 365$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.8] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3094, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{119*\x} + \tkzFct[domain=0:26, color=red, very thick]{93*\x+365} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $14.038461538461538$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 377m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 180m$ + \item $CD = 19m$ + \item $AE' = 31.3m$ + \item $AE = 93.9m$ + \item $AF = 88.2m$ + \item $E'F' = 64.03m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 180^2 + 19^2 \\ + AD^2 &= 32400 + 361 \\ + AD^2 &= 32761 \\ + AD &= \sqrt{32761} = 181m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 181 + 180 + 19 = 380m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 93.9 & AF = 88.2 & EF \\ + \hline + Triangle AE'F' & AE' = 31.3 & AF' & E'F' = 64.03 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{64.03 \times 93.9}{31.3} = 192.1 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 88.2 + 93.9 + 192.1 = 374.2m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 377m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 7 boules bleu, 9 boules jaunes, 5 boules vertes et 10 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{7}{31} \approx 0.23$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{16}{31} \approx 0.52$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{5}{31} \approx 0.16$ + + Boules rouges: $\dfrac{10}{31} \approx 0.32$ + + + Une boule rouge + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + J \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}B \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 3 & 2 & 1 & 8 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{1}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/15_DM_18_05_03_302.tex b/3e/DM/DM_Paques/15_DM_18_05_03_302.tex new file mode 100644 index 0000000..722f44d --- /dev/null +++ b/3e/DM/DM_Paques/15_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{15} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 151 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 493 \euro pour le groupe puis 118 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1510 & 3775 \\ + \hline + Prix au tarif Groupe en \euro & 493 & 1673 & 3443\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 151x$ + \item Tarif Groupe: $g: x \mapsto 118x + 493$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.6] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3926, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{151*\x} + \tkzFct[domain=0:26, color=red, very thick]{118*\x+493} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $14.93939393939394$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 10km. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 4km$ + \item $CD = 3km$ + \item $AE' = 1.9km$ + \item $AE = 3.8km$ + \item $AF = 2.2km$ + \item $E'F' = -0.3km$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 4^2 + 3^2 \\ + AD^2 &= 16 + 9 \\ + AD^2 &= 25 \\ + AD &= \sqrt{25} = 5km + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 5 + 4 + 3 = 12km + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 3.8 & AF = 2.2 & EF \\ + \hline + Triangle AE'F' & AE' = 1.9 & AF' & E'F' = -0.3 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{-0.3 \times 3.8}{1.9} = -0.6 km + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 2.2 + 3.8 + -0.6 = 5.4km + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $ACDA$ car sa longueur est plus proche de 10km. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 7 boules bleu, 7 boules jaunes, 8 boules vertes et 6 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{7}{28} \approx 0.25$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{14}{28} \approx 0.5$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{8}{28} \approx 0.29$ + + Boules rouges: $\dfrac{6}{28} \approx 0.21$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + R \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}J \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 4 & 4 & 3 & 3 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{3}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/16_DM_18_05_03_302.tex b/3e/DM/DM_Paques/16_DM_18_05_03_302.tex new file mode 100644 index 0000000..d75579f --- /dev/null +++ b/3e/DM/DM_Paques/16_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{16} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 136 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 496 \euro pour le groupe puis 97 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1360 & 3400 \\ + \hline + Prix au tarif Groupe en \euro & 496 & 1466 & 2921\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 136x$ + \item Tarif Groupe: $g: x \mapsto 97x + 496$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.7] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3536, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{136*\x} + \tkzFct[domain=0:26, color=red, very thick]{97*\x+496} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $12.717948717948717$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 287m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 126m$ + \item $CD = 32m$ + \item $AE' = 13.43m$ + \item $AE = 53.7m$ + \item $AF = 1.3m$ + \item $E'F' = 56.95m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 126^2 + 32^2 \\ + AD^2 &= 15876 + 1024 \\ + AD^2 &= 16900 \\ + AD &= \sqrt{16900} = 130m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 130 + 126 + 32 = 288m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 53.7 & AF = 1.3 & EF \\ + \hline + Triangle AE'F' & AE' = 13.43 & AF' & E'F' = 56.95 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{56.95 \times 53.7}{13.43} = 227.8 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 1.3 + 53.7 + 227.8 = 282.8m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $ACDA$ car sa longueur est plus proche de 287m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 3 boules bleu, 9 boules jaunes, 6 boules vertes et 10 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{3}{28} \approx 0.11$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{12}{28} \approx 0.43$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{6}{28} \approx 0.21$ + + Boules rouges: $\dfrac{10}{28} \approx 0.36$ + + + Une boule rouge + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + R \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}V \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & 7 & 1 & 6 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{1}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/17_DM_18_05_03_302.tex b/3e/DM/DM_Paques/17_DM_18_05_03_302.tex new file mode 100644 index 0000000..7952d63 --- /dev/null +++ b/3e/DM/DM_Paques/17_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{17} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 163 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 521 \euro pour le groupe puis 121 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1630 & 4075 \\ + \hline + Prix au tarif Groupe en \euro & 521 & 1731 & 3546\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 163x$ + \item Tarif Groupe: $g: x \mapsto 121x + 521$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.6] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=4238, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{163*\x} + \tkzFct[domain=0:26, color=red, very thick]{121*\x+521} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $12.404761904761905$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 117m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 48m$ + \item $CD = 20m$ + \item $AE' = 3.88m$ + \item $AE = 15.5m$ + \item $AF = 37.3m$ + \item $E'F' = 14.8m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 48^2 + 20^2 \\ + AD^2 &= 2304 + 400 \\ + AD^2 &= 2704 \\ + AD &= \sqrt{2704} = 52m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 52 + 48 + 20 = 120m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 15.5 & AF = 37.3 & EF \\ + \hline + Triangle AE'F' & AE' = 3.88 & AF' & E'F' = 14.8 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{14.8 \times 15.5}{3.88} = 59.2 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 37.3 + 15.5 + 59.2 = 112.0m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $ACDA$ car sa longueur est plus proche de 117m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 10 boules bleu, 7 boules jaunes, 4 boules vertes et 4 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{10}{25} \approx 0.4$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{17}{25} \approx 0.68$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{4}{25} \approx 0.16$ + + Boules rouges: $\dfrac{4}{25} \approx 0.16$ + + + Une boule rouge + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + V \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}B \hspace{0.4cm}V \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}J \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 5 & 4 & 5 & \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{5}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/18_DM_18_05_03_302.tex b/3e/DM/DM_Paques/18_DM_18_05_03_302.tex new file mode 100644 index 0000000..f6e87f9 --- /dev/null +++ b/3e/DM/DM_Paques/18_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{18} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 125 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 377 \euro pour le groupe puis 92 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1250 & 3125 \\ + \hline + Prix au tarif Groupe en \euro & 377 & 1297 & 2677\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 125x$ + \item Tarif Groupe: $g: x \mapsto 92x + 377$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.7] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3250, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{125*\x} + \tkzFct[domain=0:26, color=red, very thick]{92*\x+377} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $11.424242424242424$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 260m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 60m$ + \item $CD = 91m$ + \item $AE' = 53.45m$ + \item $AE = 106.9m$ + \item $AF = 23.3m$ + \item $E'F' = 62.6m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 60^2 + 91^2 \\ + AD^2 &= 3600 + 8281 \\ + AD^2 &= 11881 \\ + AD &= \sqrt{11881} = 109m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 109 + 60 + 91 = 260m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 106.9 & AF = 23.3 & EF \\ + \hline + Triangle AE'F' & AE' = 53.45 & AF' & E'F' = 62.6 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{62.6 \times 106.9}{53.45} = 125.2 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 23.3 + 106.9 + 125.2 = 255.4m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $ACDA$ car sa longueur est plus proche de 260m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 2 boules bleu, 7 boules jaunes, 6 boules vertes et 10 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{2}{25} \approx 0.08$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{9}{25} \approx 0.36$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{6}{25} \approx 0.24$ + + Boules rouges: $\dfrac{10}{25} \approx 0.4$ + + + Une boule rouge + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + R \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}R \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 2 & 4 & & 8 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{0}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/19_DM_18_05_03_302.tex b/3e/DM/DM_Paques/19_DM_18_05_03_302.tex new file mode 100644 index 0000000..fb045ad --- /dev/null +++ b/3e/DM/DM_Paques/19_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{19} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 100 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 367 \euro pour le groupe puis 79 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1000 & 2500 \\ + \hline + Prix au tarif Groupe en \euro & 367 & 1157 & 2342\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 100x$ + \item Tarif Groupe: $g: x \mapsto 79x + 367$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.9] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=2600, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{100*\x} + \tkzFct[domain=0:26, color=red, very thick]{79*\x+367} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $17.476190476190474$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 84km. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 12km$ + \item $CD = 35km$ + \item $AE' = 17.15km$ + \item $AE = 34.3km$ + \item $AF = 37.1km$ + \item $E'F' = 3.1km$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 12^2 + 35^2 \\ + AD^2 &= 144 + 1225 \\ + AD^2 &= 1369 \\ + AD &= \sqrt{1369} = 37km + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 37 + 12 + 35 = 84km + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 34.3 & AF = 37.1 & EF \\ + \hline + Triangle AE'F' & AE' = 17.15 & AF' & E'F' = 3.1 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{3.1 \times 34.3}{17.15} = 6.2 km + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 37.1 + 34.3 + 6.2 = 77.6km + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $ACDA$ car sa longueur est plus proche de 84km. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 2 boules bleu, 10 boules jaunes, 9 boules vertes et 8 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{2}{29} \approx 0.07$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{12}{29} \approx 0.41$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{9}{29} \approx 0.31$ + + Boules rouges: $\dfrac{8}{29} \approx 0.28$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + V \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}V \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 1 & 4 & 8 & 1 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{8}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/20_DM_18_05_03_302.tex b/3e/DM/DM_Paques/20_DM_18_05_03_302.tex new file mode 100644 index 0000000..4c63670 --- /dev/null +++ b/3e/DM/DM_Paques/20_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{20} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 135 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 417 \euro pour le groupe puis 107 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1350 & 3375 \\ + \hline + Prix au tarif Groupe en \euro & 417 & 1487 & 3092\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 135x$ + \item Tarif Groupe: $g: x \mapsto 107x + 417$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.7] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3510, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{135*\x} + \tkzFct[domain=0:26, color=red, very thick]{107*\x+417} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $14.892857142857142$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 52km. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 24km$ + \item $CD = 7km$ + \item $AE' = 3.15km$ + \item $AE = 6.3km$ + \item $AF = 12.3km$ + \item $E'F' = 16.2km$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 24^2 + 7^2 \\ + AD^2 &= 576 + 49 \\ + AD^2 &= 625 \\ + AD &= \sqrt{625} = 25km + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 25 + 24 + 7 = 56km + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 6.3 & AF = 12.3 & EF \\ + \hline + Triangle AE'F' & AE' = 3.15 & AF' & E'F' = 16.2 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{16.2 \times 6.3}{3.15} = 32.4 km + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 12.3 + 6.3 + 32.4 = 51.0km + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 52km. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 5 boules bleu, 8 boules jaunes, 3 boules vertes et 3 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{5}{19} \approx 0.26$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{13}{19} \approx 0.68$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{3}{19} \approx 0.16$ + + Boules rouges: $\dfrac{3}{19} \approx 0.16$ + + + Une boule rouge + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + J \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}B \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 2 & 7 & 3 & 2 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{3}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/21_DM_18_05_03_302.tex b/3e/DM/DM_Paques/21_DM_18_05_03_302.tex new file mode 100644 index 0000000..ef9315f --- /dev/null +++ b/3e/DM/DM_Paques/21_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{21} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 128 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 426 \euro pour le groupe puis 98 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1280 & 3200 \\ + \hline + Prix au tarif Groupe en \euro & 426 & 1406 & 2876\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 128x$ + \item Tarif Groupe: $g: x \mapsto 98x + 426$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.7] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3328, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{128*\x} + \tkzFct[domain=0:26, color=red, very thick]{98*\x+426} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $14.2$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 217m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 20m$ + \item $CD = 99m$ + \item $AE' = 26.88m$ + \item $AE = 107.5m$ + \item $AF = 67.9m$ + \item $E'F' = 9.7m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 20^2 + 99^2 \\ + AD^2 &= 400 + 9801 \\ + AD^2 &= 10201 \\ + AD &= \sqrt{10201} = 101m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 101 + 20 + 99 = 220m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 107.5 & AF = 67.9 & EF \\ + \hline + Triangle AE'F' & AE' = 26.88 & AF' & E'F' = 9.7 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{9.7 \times 107.5}{26.88} = 38.8 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 67.9 + 107.5 + 38.8 = 214.2m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 217m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 2 boules bleu, 3 boules jaunes, 6 boules vertes et 7 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{2}{18} \approx 0.11$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{5}{18} \approx 0.28$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{6}{18} \approx 0.33$ + + Boules rouges: $\dfrac{7}{18} \approx 0.39$ + + + Une boule rouge + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + R \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}R \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 2 & 3 & 2 & 7 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{2}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/22_DM_18_05_03_302.tex b/3e/DM/DM_Paques/22_DM_18_05_03_302.tex new file mode 100644 index 0000000..a9afa1a --- /dev/null +++ b/3e/DM/DM_Paques/22_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{22} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 149 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 489 \euro pour le groupe puis 108 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1490 & 3725 \\ + \hline + Prix au tarif Groupe en \euro & 489 & 1569 & 3189\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 149x$ + \item Tarif Groupe: $g: x \mapsto 108x + 489$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.6] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3874, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{149*\x} + \tkzFct[domain=0:26, color=red, very thick]{108*\x+489} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $11.926829268292684$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 373m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 180m$ + \item $CD = 19m$ + \item $AE' = 70.6m$ + \item $AE = 141.2m$ + \item $AF = 52.9m$ + \item $E'F' = 89.85m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 180^2 + 19^2 \\ + AD^2 &= 32400 + 361 \\ + AD^2 &= 32761 \\ + AD &= \sqrt{32761} = 181m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 181 + 180 + 19 = 380m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 141.2 & AF = 52.9 & EF \\ + \hline + Triangle AE'F' & AE' = 70.6 & AF' & E'F' = 89.85 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{89.85 \times 141.2}{70.6} = 179.7 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 52.9 + 141.2 + 179.7 = 373.8m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 373m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 6 boules bleu, 3 boules jaunes, 8 boules vertes et 7 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{6}{24} \approx 0.25$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{9}{24} \approx 0.38$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{8}{24} \approx 0.33$ + + Boules rouges: $\dfrac{7}{24} \approx 0.29$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + V \hspace{0.4cm}B \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}V \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 3 & 1 & 7 & 3 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{7}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/23_DM_18_05_03_302.tex b/3e/DM/DM_Paques/23_DM_18_05_03_302.tex new file mode 100644 index 0000000..68d32af --- /dev/null +++ b/3e/DM/DM_Paques/23_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{23} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 173 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 571 \euro pour le groupe puis 137 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1730 & 4325 \\ + \hline + Prix au tarif Groupe en \euro & 571 & 1941 & 3996\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 173x$ + \item Tarif Groupe: $g: x \mapsto 137x + 571$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.5] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=4498, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{173*\x} + \tkzFct[domain=0:26, color=red, very thick]{137*\x+571} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $15.86111111111111$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 125m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 28m$ + \item $CD = 45m$ + \item $AE' = 12.5m$ + \item $AE = 25.0m$ + \item $AF = 57.2m$ + \item $E'F' = 19.4m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 28^2 + 45^2 \\ + AD^2 &= 784 + 2025 \\ + AD^2 &= 2809 \\ + AD &= \sqrt{2809} = 53m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 53 + 28 + 45 = 126m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 25.0 & AF = 57.2 & EF \\ + \hline + Triangle AE'F' & AE' = 12.5 & AF' & E'F' = 19.4 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{19.4 \times 25.0}{12.5} = 38.8 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 57.2 + 25.0 + 38.8 = 121.0m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $ACDA$ car sa longueur est plus proche de 125m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 3 boules bleu, 9 boules jaunes, 4 boules vertes et 7 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{3}{23} \approx 0.13$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{12}{23} \approx 0.52$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{4}{23} \approx 0.17$ + + Boules rouges: $\dfrac{7}{23} \approx 0.3$ + + + Une boule rouge + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + R \hspace{0.4cm}R \hspace{0.4cm}B \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}B \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}J \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 2 & 4 & 1 & 7 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{1}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/24_DM_18_05_03_302.tex b/3e/DM/DM_Paques/24_DM_18_05_03_302.tex new file mode 100644 index 0000000..648018f --- /dev/null +++ b/3e/DM/DM_Paques/24_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{24} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 118 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 444 \euro pour le groupe puis 84 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1180 & 2950 \\ + \hline + Prix au tarif Groupe en \euro & 444 & 1284 & 2544\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 118x$ + \item Tarif Groupe: $g: x \mapsto 84x + 444$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.8] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3068, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{118*\x} + \tkzFct[domain=0:26, color=red, very thick]{84*\x+444} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $13.058823529411764$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 96km. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 24km$ + \item $CD = 32km$ + \item $AE' = 6.65km$ + \item $AE = 26.6km$ + \item $AF = 13.7km$ + \item $E'F' = 12.18km$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 24^2 + 32^2 \\ + AD^2 &= 576 + 1024 \\ + AD^2 &= 1600 \\ + AD &= \sqrt{1600} = 40km + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 40 + 24 + 32 = 96km + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 26.6 & AF = 13.7 & EF \\ + \hline + Triangle AE'F' & AE' = 6.65 & AF' & E'F' = 12.18 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{12.18 \times 26.6}{6.65} = 48.7 km + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 13.7 + 26.6 + 48.7 = 89.0km + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $ACDA$ car sa longueur est plus proche de 96km. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 3 boules bleu, 8 boules jaunes, 9 boules vertes et 7 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{3}{27} \approx 0.11$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{11}{27} \approx 0.41$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{9}{27} \approx 0.33$ + + Boules rouges: $\dfrac{7}{27} \approx 0.26$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + V \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}R \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & 5 & 5 & 4 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{5}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/25_DM_18_05_03_302.tex b/3e/DM/DM_Paques/25_DM_18_05_03_302.tex new file mode 100644 index 0000000..5be349e --- /dev/null +++ b/3e/DM/DM_Paques/25_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{25} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 148 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 589 \euro pour le groupe puis 114 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1480 & 3700 \\ + \hline + Prix au tarif Groupe en \euro & 589 & 1729 & 3439\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 148x$ + \item Tarif Groupe: $g: x \mapsto 114x + 589$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.6] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3848, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{148*\x} + \tkzFct[domain=0:26, color=red, very thick]{114*\x+589} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $17.323529411764707$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 286m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 126m$ + \item $CD = 32m$ + \item $AE' = 21.4m$ + \item $AE = 42.8m$ + \item $AF = 120.5m$ + \item $E'F' = 59.95m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 126^2 + 32^2 \\ + AD^2 &= 15876 + 1024 \\ + AD^2 &= 16900 \\ + AD &= \sqrt{16900} = 130m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 130 + 126 + 32 = 288m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 42.8 & AF = 120.5 & EF \\ + \hline + Triangle AE'F' & AE' = 21.4 & AF' & E'F' = 59.95 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{59.95 \times 42.8}{21.4} = 119.9 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 120.5 + 42.8 + 119.9 = 283.2m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $ACDA$ car sa longueur est plus proche de 286m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 8 boules bleu, 6 boules jaunes, 6 boules vertes et 2 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{8}{22} \approx 0.36$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{14}{22} \approx 0.64$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{6}{22} \approx 0.27$ + + Boules rouges: $\dfrac{2}{22} \approx 0.09$ + + + Une boule verte + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + B \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}B \hspace{0.4cm}V \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 6 & 4 & 4 & \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{4}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/26_DM_18_05_03_302.tex b/3e/DM/DM_Paques/26_DM_18_05_03_302.tex new file mode 100644 index 0000000..30bfbd7 --- /dev/null +++ b/3e/DM/DM_Paques/26_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{26} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 113 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 449 \euro pour le groupe puis 80 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1130 & 2825 \\ + \hline + Prix au tarif Groupe en \euro & 449 & 1249 & 2449\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 113x$ + \item Tarif Groupe: $g: x \mapsto 80x + 449$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.8] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=2938, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{113*\x} + \tkzFct[domain=0:26, color=red, very thick]{80*\x+449} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $13.606060606060606$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 237m. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 40m$ + \item $CD = 96m$ + \item $AE' = 23.17m$ + \item $AE = 69.5m$ + \item $AF = 2.2m$ + \item $E'F' = 54.57m$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 40^2 + 96^2 \\ + AD^2 &= 1600 + 9216 \\ + AD^2 &= 10816 \\ + AD &= \sqrt{10816} = 104m + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 104 + 40 + 96 = 240m + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 69.5 & AF = 2.2 & EF \\ + \hline + Triangle AE'F' & AE' = 23.17 & AF' & E'F' = 54.57 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{54.57 \times 69.5}{23.17} = 163.7 m + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 2.2 + 69.5 + 163.7 = 235.4m + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 237m. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 2 boules bleu, 5 boules jaunes, 3 boules vertes et 6 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{2}{16} \approx 0.12$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{7}{16} \approx 0.44$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{3}{16} \approx 0.19$ + + Boules rouges: $\dfrac{6}{16} \approx 0.38$ + + + Une boule rouge + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + B \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}B \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}R \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 2 & 7 & 3 & 2 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{3}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/27_DM_18_05_03_302.tex b/3e/DM/DM_Paques/27_DM_18_05_03_302.tex new file mode 100644 index 0000000..f2e0d6d --- /dev/null +++ b/3e/DM/DM_Paques/27_DM_18_05_03_302.tex @@ -0,0 +1,278 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{27} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + % theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: 146 \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de 494 \euro pour le groupe puis 111 \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & 1460 & 3650 \\ + \hline + Prix au tarif Groupe en \euro & 494 & 1604 & 3269\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto 146x$ + \item Tarif Groupe: $g: x \mapsto 111x + 494$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + % On force que le graphique soit légèrement plus grand que 12 cm + \begin{tikzpicture}[yscale=0.6] + \tkzInit[xmin=0,xmax=26, + ymin=0,ymax=3796, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:26, color=blue, very thick]{146*\x} + \tkzFct[domain=0:26, color=red, very thick]{111*\x+494} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $14.114285714285714$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + % exo de geometrie comme au brevet blanc. + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de 62km. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = 20km$ + \item $CD = 21km$ + \item $AE' = 1.2km$ + \item $AE = 3.6km$ + \item $AF = 8.3km$ + \item $E'F' = 16.77km$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= 20^2 + 21^2 \\ + AD^2 &= 400 + 441 \\ + AD^2 &= 841 \\ + AD &= \sqrt{841} = 29km + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = 29 + 20 + 21 = 70km + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = 3.6 & AF = 8.3 & EF \\ + \hline + Triangle AE'F' & AE' = 1.2 & AF' & E'F' = 16.77 \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{16.77 \times 3.6}{1.2} = 50.3 km + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = 8.3 + 3.6 + 50.3 = 62.2km + \end{align*} + + \item Choix du parcours: + + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de 62km. + \end{itemize} +\end{solution} + +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \renewcommand{\arraystretch}{1.5} + + + + + + + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a 4 boules bleu, 6 boules jaunes, 7 boules vertes et 9 boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{4}{26} \approx 0.15$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{10}{26} \approx 0.38$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{7}{26} \approx 0.27$ + + Boules rouges: $\dfrac{9}{26} \approx 0.35$ + + + Une boule rouge + + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + + + + \begin{center} + B \hspace{0.4cm}R \hspace{0.4cm}V \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}J \hspace{0.4cm}J \hspace{0.4cm}V \hspace{0.4cm}V \hspace{0.4cm}J \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}R \hspace{0.4cm}R \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & 1 & 3 & 3 & 7 \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{3}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + + + \item Combien de points a-t-on gagné au total? + \begin{solution} + 17 + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + 4.25 + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + 3.5 + \end{solution} + + \end{enumerate} +\end{enumerate} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/3e/DM/DM_Paques/Proba_stat.tex b/3e/DM/DM_Paques/Proba_stat.tex new file mode 100644 index 0000000..11a2dcb --- /dev/null +++ b/3e/DM/DM_Paques/Proba_stat.tex @@ -0,0 +1,100 @@ +\renewcommand{\arraystretch}{1.5} + +\Block{set bBleu = randint(2,10)} +\Block{set bJaune = randint(2,10)} +\Block{set bVerte = randint(2,10)} +\Block{set bRouge = randint(2,10)} +\Block{set nbrTot = bBleu + bJaune + bVerte + bRouge} + +Dans une urne, on a placé des boules colorées indiscernables au touché. Il y a \Var{bBleu} boules bleu, \Var{bJaune} boules jaunes, \Var{bVerte} boules vertes et \Var{bRouge} boules rouges. + +\begin{enumerate} + \item % Proba + \begin{enumerate} + \item Quelle est la probabilité de tirer une boule bleu? + \begin{solution} + $\dfrac{\Var{bBleu}}{\Var{nbrTot}} \approx \Var{(bBleu / nbrTot) |round(2)}$ + \end{solution} + \item Quelle est la probabilté de tirer une boule jaune ou bleu? + \begin{solution} + $\dfrac{\Var{bJaune + bBleu}}{\Var{nbrTot}} \approx \Var{((bJaune + bBleu)/ nbrTot) | round(2)}$ + \end{solution} + \item A-t-on plus de chance de tirer une boule verte ou une boule rouge? + \begin{solution} + Boules vertes: $\dfrac{\Var{bVerte}}{\Var{nbrTot}} \approx \Var{(bVerte / nbrTot) |round(2)}$ + + Boules rouges: $\dfrac{\Var{bRouge}}{\Var{nbrTot}} \approx \Var{(bRouge / nbrTot) |round(2)}$ + + \Block{if bVerte > bRouge} + Une boule verte + \Block{else} + Une boule rouge + \Block{endif} + \end{solution} + \end{enumerate} + + \item % Stat + On effectue 14 tirages (avec remise) dans cette urne et on obtient les couleurs suivantes: + + \Block{set urne = ['B']*bBleu + + ['J']*bJaune + + ['V']*bVerte + + ['R']*bRouge + } + \Block{set sample = Dataset.random(14, distrib="choice", rd_args=[urne], nbr_format=str)} + + \begin{center} + \Var{sample | join(" \hspace{0.4cm}")} \\ + \end{center} + + \begin{enumerate} + \item Compléter le tableau des effectifs ci-dessous + + \Block{set wsample = WeightedDataset(sample)} + + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & & & & \\ + \hline + \end{tabular} + \begin{solution} + \begin{tabular}{|c|*{4}{c|}} + \hline + Couleur & Bleu & Jaune & Vert & Rouge \\ + \hline + Effectif & \Var{wsample['B']} & \Var{wsample['J']} & \Var{wsample['V']} & \Var{wsample['R']} \\ + \hline + \end{tabular} + \end{solution} + + \item Calculer la fréquence des boules vertes. + + \begin{solution} + Fréquence de boules vertes: $\frac{\Var{sample.count('V')}}{14}$ + \end{solution} + + \end{enumerate} + + \item À chaque couleur, on associe des points. Une boule bleu rapporte 10 points, une boule jaune 5 points, une boule verte 2 points et une boule rouge 0 points. + \begin{enumerate} + + + \Block{set wPts = WeightedDataset([10, 5, 2, 0], [wsample['B'], wsample['J'], wsample['V'], wsample['R']])} + + \item Combien de points a-t-on gagné au total? + \begin{solution} + \Var{wPts.sum()} + \end{solution} + \item Calculer la moyenne des gains. + \begin{solution} + \Var{wPts.mean()} + \end{solution} + \item Calculer la médiane des gains. + \begin{solution} + \Var{wPts.quartile(2)} + \end{solution} + + \end{enumerate} +\end{enumerate} diff --git a/3e/DM/DM_Paques/all_DM_18_05_03_302.pdf b/3e/DM/DM_Paques/all_DM_18_05_03_302.pdf new file mode 100644 index 0000000000000000000000000000000000000000..230b43410be457a0d921aa18f854e310d49688b1 GIT binary patch literal 453377 zcmd43WpEummo;i;=9oFAnBkb2nK@>rV}_V1W@culn3*Alm>n~-9W(RiJTUj2`JSqG zZrz%x>mOa(Y8`3s-Q8trttC+?h=_e+qGyAnn4BG0fCI1*GZNbwTEg-0z%j^}*qS++ 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zUr07NM3F?Kh&98YNgN#@9f^Mr!{3vV1V6v$nl9Rb zjY#N;nG6Uc5okOX084^pBw>KCvzHg{oD`AL5`|891kM?a^pwQnokYH;tsx%k;Ef~_ z|NmU=FM139`(Fv)j}O8lw$mkdOTy?@D|5Te!oDT6}fh)eya3(II|X;~PpprD3 + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + + A + C + D + E + F + F' + E' + Départ et arrivée + + + + diff --git a/3e/DM/DM_Paques/parcours.tex b/3e/DM/DM_Paques/parcours.tex new file mode 100644 index 0000000..bfe53e8 --- /dev/null +++ b/3e/DM/DM_Paques/parcours.tex @@ -0,0 +1,91 @@ +% exo de geometrie comme au brevet blanc. + +%- set AD, AC, DC = random_pythagore() +%- set tourACDA = AC+AD+DC +%- set AE, AF = round(tourACDA/2*random(), 1), round(tourACDA/2*random(), 1) +%- set EF = round(tourACDA - AE - AF - randint(20,40)*0.2, 1) +%- set tourAEFA = round(AE+EF+AF, 1) +%- set rapport = randint(2,5) +%- set AE1, AF1, EF1 = round(AE/rapport,2) , round(AF/rapport,2), round(EF/rapport,2) +%- set objectif = randint(floor(tourAEFA), tourACDA) +%- if objectif > 100 + %- set unit = "m" +%- else + %- set unit = "km" +%- endif + + +Une commune souhaite aménager des parcours de santé sur son territoire. On fait deux propositions au conseil municipale, schématisés ci-dessous: +\begin{itemize} + \item Le parcours ACDA + \item Le parcours AEFA +\end{itemize} +Ils souhaitent faire un parcours dont la longueur s'approche le plus possible de \Var{objectif}\Var{unit}. + +Peux-tu les aider à choisir le parcours? Justifie + +\textbf{Attention: La figure proposée au conseil municipale n'est pas à l'échelle, mais les codages et les dimension données sont correctes.} + +\begin{minipage}{0.6\textwidth} + \includegraphics[scale = 0.4]{./fig/parcours} +\end{minipage} +\begin{minipage}{0.4\textwidth} + \begin{itemize} + \item $AC = \Var{AC}\Var{unit}$ + \item $CD = \Var{DC}\Var{unit}$ + \item $AE' = \Var{AE1}\Var{unit}$ + \item $AE = \Var{AE}\Var{unit}$ + \item $AF = \Var{AF}\Var{unit}$ + \item $E'F' = \Var{EF1}\Var{unit}$ + \item $(E'F') // (EF)$ + \item L'angle $\widehat{EAF}$ vaut $30^o$ + \end{itemize} +\end{minipage} +\begin{solution} + \begin{itemize} + \item Parcours ACDA: + + D'après la figure, on voit que le triangle $ACD$ est rectangle en $C$ donc d'après le théorème de Pythagore, on a + \begin{align*} + AD^2 &= AC^2 + DC^2 \\ + AD^2 &= \Var{AC}^2 + \Var{DC}^2 \\ + AD^2 &= \Var{AC**2} + \Var{DC**2} \\ + AD^2 &= \Var{AC**2 + DC**2} \\ + AD &= \sqrt{\Var{AC**2 + DC**2}} = \Var{AD}\Var{unit} + \end{align*} + Donc le parcours ACDA mesure + \begin{align*} + AD + AC + CD = \Var{AD} + \Var{AC} + \Var{DC} = \Var{tourACDA}\Var{unit} + \end{align*} + + \item Parcours AEFA: + + D'après les données, on sait que $(EF) // (E'F')$. On voit aussi que $A$, $E'$ et $E$ sont alignés. Il en est de même pour les points $A$, $F'$ et $F$. Donc d'après le théorème de Thalès + + \begin{tabular}{|c|c|c|c|} + \hline + Triangle AEF & AE = \Var{AE} & AF = \Var{AF} & EF \\ + \hline + Triangle AE'F' & AE' = \Var{AE1} & AF' & E'F' = \Var{EF1} \\ + \hline + \end{tabular} + est un tableau de proportionnalité. Donc on peut faire un produit en croix pour calcul $EF$. + \begin{align*} + EF = \frac{E'F' \times AE}{AE'} = \frac{\Var{EF1} \times \Var{AE}}{\Var{AE1}} = \Var{EF} \Var{unit} + \end{align*} + + Donc le parcours AEFA mesure + \begin{align*} + AF + AE + EF = \Var{AF} + \Var{AE} + \Var{EF} = \Var{tourAEFA}\Var{unit} + \end{align*} + + \item Choix du parcours: + + %- if abs(tourACDA - objectif) < abs(tourAEFA - objectif) + Il faudra choisir le tour $ACDA$ car sa longueur est plus proche de \Var{objectif}\Var{unit}. + %- else + Il faudra choisir le tour $AFEA$ car sa longueur est plus proche de \Var{objectif}\Var{unit}. + %- endif + \end{itemize} +\end{solution} + diff --git a/3e/DM/DM_Paques/tarif_danseurs.tex b/3e/DM/DM_Paques/tarif_danseurs.tex new file mode 100644 index 0000000..87a2b54 --- /dev/null +++ b/3e/DM/DM_Paques/tarif_danseurs.tex @@ -0,0 +1,82 @@ +% theme: Fonction linéaire, Fonction affine +% require: tkz-fct + + +%- set I_a = randint(100, 200) +%- set G_b =randint(3*I_a, 4*I_a) +%- set G_a = randint(I_a*14//20, I_a*16//20) + +Taraina veut inscrire ses 21 élèves à un entrainement pour l'évènement \textbf{Battle of the year}. + +Deux tarifs lui sont proposés: +\begin{itemize} + \item Tarif Individuel: \Var{I_a} \euro par danseur inscrit. + \item Tarif Groupe : Paiement d'un forfait de \Var{G_b} \euro pour le groupe puis \Var{G_a} \euro par danseur inscrit. +\end{itemize} + +\medskip + +\begin{enumerate} +\item Complète le tableau suivant : + +\medskip + +\begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & & & \\ + \hline + Prix au tarif Groupe en \euro & & & \\ + \hline +\end{tabularx} + \begin{solution} + \begin{tabularx}{0.7\linewidth}{|c|*{3}{>{\centering \arraybackslash}X|}}\hline + Nombre d'inscriptions & 0 & 10 & 25\\ + \hline + Prix au tarif Individuel en \euro & 0 & \Var{10*I_a} & \Var{25*I_a} \\ + \hline + Prix au tarif Groupe en \euro & \Var{G_b} & \Var{G_b + 10*G_a} & \Var{G_b + 25*G_a}\\ + \hline + \end{tabularx} + \end{solution} + +\medskip + +\item Pour chacun des tarifs, exprimer le prix en fonction du nombre de danseurs inscrits. + + \begin{solution} + $x$ représente ici le nombre d'élèves inscrits. + \begin{itemize} + \item Tarif Individuel: $f: x \mapsto \Var{I_a}x$ + \item Tarif Groupe: $g: x \mapsto \Var{G_a}x + \Var{G_b}$ + \end{itemize} + \end{solution} + + +\item Tracer sur le graphique suivant, les courbes représentants les 2 tarifs proposés. + + %- set xmax = 26 + %- set ymax = I_a*26 + % On force que le graphique soit légèrement plus grand que 12 cm + %- set yscale = round(12*200/ymax, 1) + + \begin{tikzpicture}[yscale=\Var{yscale}] + \tkzInit[xmin=0,xmax=\Var{xmax}, + ymin=0,ymax=\Var{ymax}, + xstep=2,ystep=200] + \tkzAxeX[thick, poslabel=right,label=] + \tkzAxeY[thick, poslabel=above,label=] + \tkzDrawX[label={\textit{Danseurs inscrits}},below= -12pt] + \tkzDrawY[label={\textit{Prix}}, below=-10pt] + \tkzGrid + \tkzFct[domain=0:\Var{xmax}, color=blue, very thick]{\Var{I_a}*\x} + \tkzFct[domain=0:\Var{xmax}, color=red, very thick]{\Var{G_a}*\x+\Var{G_b}} + \end{tikzpicture} + +\item Pour quel nombre d'inscriptions paye-t-on le même prix quel que soit le tarif choisi? + \begin{solution} + $\Var{G_b/(I_a - G_a)}$ % est entre 10 et 20 par contruction des paramètres + \end{solution} +\end{enumerate} + + diff --git a/3e/DM/DM_Paques/tpl_DM_18_05_03_302.tex b/3e/DM/DM_Paques/tpl_DM_18_05_03_302.tex new file mode 100644 index 0000000..c99bd1f --- /dev/null +++ b/3e/DM/DM_Paques/tpl_DM_18_05_03_302.tex @@ -0,0 +1,42 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tkz-fct} + +\usepackage{wrapfig} + +\title{DM de Paques} +\tribe{302} +\date{Jeudi 3 mai 2018} + +\sujet{\Var{infos.num}} + +%\geometry{left=10mm,right=10mm, bottom= 10mm, top=10mm} +\xsimsetup{ +solution/print = false +} + +\begin{document} + +\maketitle +\bigskip +{\Large \textbf{Nom - Prénom:}} + +\begin{exercise}[subtitle={Battle of the year}] + \Block{include "./tarif_danseurs.tex"} +\end{exercise} + +\begin{exercise}[subtitle={Parcours}] + \Block{include "./parcours.tex"} +\end{exercise} + +\begin{exercise}[subtitle={Tirages au sort}] + \Block{include "./Proba_stat.tex"} +\end{exercise} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: +