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RBJ%KZb0bhviYrPW{4ZpoDgpoi diff --git a/TST/DS/DS_21_02_26/exercises.tex b/TST/DS/DS_21_02_26/exercises.tex index d6bd5d1..732d1ed 100644 --- a/TST/DS/DS_21_02_26/exercises.tex +++ b/TST/DS/DS_21_02_26/exercises.tex @@ -69,139 +69,27 @@ \end{exercise} \begin{exercise}[subtitle={Portique }, points=7, tribe={1}, type={Exercise}] - On estime que 23\% des voyageurs font sonner le portique d'un aéroport. On suppose que chaque voyageur fait sonner ce portique indépendamment des autres. + En France, la probabilité de la naissance d'une garçon est $ p = 0.515$ à chaque naissance (on ne considèrera pas les cas des jumeaux). - Arrive un groupe de 3 personnes. On s'intéresse au nombre de fois que le portique va sonner. + On choisit au hasard 3 familles avec un enfant unique et on s'intéresse au nombre de garçon. + + On note $X$ la variable aléatoire qui compte le nombre de garçons. - On note $X$ la variable aléatoire qui compte le nombre de fois que le portique sonne pour 3 personnes. \begin{enumerate} % 2pts \item Faire un arbre de probabilité modélisant cette situation. % 1pt \item Quelle loi suit $X$, préciser ses paramètres. % 1pt - \item Quelle est la probabilité que le portique sonne 2 fois? + \item Quelle est la probabilité qu'exactement 2 familles aient un garçon? % 2pts - \item Calculer les quantités suivantes + \item Calculer puis interpréter les quantités suivantes \[ P(X = 0) \qquad \qquad P(X \leq 1) \] % 1pt - \item Combien en moyenne le portique va-t-il sonner quand 3 personnes passent? + \item Combien en moyenne les 3 familles sélectionnées auront de garçons? \end{enumerate} \end{exercise} -\begin{exercise}[subtitle={Automatismes}, points=6, tribe={2}, type={automatismes}] - \begin{enumerate} - \item Simplifier le calcul suivant - \begin{bclogo}[barre=none, logo=]{Réponse} - $\ds \frac{2\times10^{4} \times 10^{-2}\times 6}{5\times 10^2 \times 10^6} =$ - \vspace{2cm} - \end{bclogo} - - \item Une quantité est augmentée de 30\%. Quel taux d'évolution doit-on appliqué pour la faire revenir à sa valeur initiale? - \reponse{2.5cm} - - \item En 2010, la chiffre d'affaire d'une entreprise était de \np{25 000}. Chaque année, il a progressé de 11\%. Quel est le taux d'évolution global entre 2010 et 2020? - \reponse{2.5cm} - - \item En 2015, j'achète une voiture \np{12000}\euro. En 2019, elle a perdu 50\% de sa valeur. Quelle a été la perte annuelle moyenne? - \reponse{2.5cm} - - \item Convertir $89,45m^3$ en $cm^3$ - \reponse{2cm} - - \pagebreak - - \item Convertir 2,75h en heure et minutes. - \reponse{2cm} - \end{enumerate} -\end{exercise} - -\begin{exercise}[subtitle={Équation et logarithme}, points=4.5, tribe={2}, type={Exercise}] - Résoudre par un calcul les équations et inéquations suivantes - \begin{multicols}{3} - \begin{enumerate} - \item $10^x = 250$ - \item $10^{-3x + 1} \leq 5$ - \item $4\times 10^x = 100$ - \end{enumerate} - \end{multicols} -\end{exercise} - -\begin{exercise}[subtitle={Production en transition}, points=9.5, tribe={2}, type={Exercise}] - \noindent - Une usine qui fabrique un produit A, décide de fabriquer un nouveau produit B afin d'augmenter son chiffre d'affaires. La quantité, exprimée en tonnes, fabriquée par jour par l'usine est modélisée par : - \begin{multicols}{2} - \begin{itemize} - \item la fonction $f$ définie sur [0~;~14] par - - \[f(x) = \np{2000}\times 0.81^{x}\] - - pour le produit A ; - \item la fonction $g$ définie sur [0~;~14] par - - \[g (x)= 15x^2 + 50 x\] - - pour le produit B - \end{itemize} - \end{multicols} - Où $x$ est la durée écoulée depuis le lancement du nouveau produit B exprimée en mois. - - \noindent - \begin{minipage}{0.5\textwidth} - \textbf{Partie A} - - Leurs courbes représentatives respectives $\mathcal{C}_f$ et $\mathcal{C}_g$ sont données ci-contre. - - Par lecture graphique, sans justification et avec la précision permise par le graphique : - - - \begin{enumerate} - \item Quelle est la quantité de produit A au lancement du produit B? - \item Quelle est la quantité de produit B produite 9 mois après le lancement? - \item Déterminer la durée nécessaire pour que la quantité de produit B dépasse celle du produit A. - \item L'usine ne peut pas fabriquer une quantité journalière de produit B supérieure à \np{3000}~tonnes. - - Au bout de combien de mois cette quantité journalière sera atteinte? - \end{enumerate} - \end{minipage} - \begin{minipage}{0.5\textwidth} - \begin{tikzpicture}[xscale=0.6, yscale=1] - \tkzInit[xmin=0,xmax=14,xstep=1, - ymin=0,ymax=3500,ystep=500] - \tkzGrid - \tkzGrid[sub, subystep=100, subxstep=1] - \tkzDrawX[label={\textit{Temps (en mois)}},below=10pt] - \tkzLabelX - \tkzDrawY[label={\textit{Production (en tonnes)}}, right=10pt] - \tkzLabelY - - \tkzFct[domain=0:14,color=red,very thick]{2000*0.81**(\x)} - \tkzDefPointByFct(1) - \tkzText[above right,text=red](tkzPointResult){${\mathcal{C}}_f$} - - \tkzFct[domain=0:14,color=black,very thick]{15*\x**2 + 50*\x} - \tkzDefPointByFct(1) - \tkzText[above right,text=black](tkzPointResult){${\mathcal{C}}_g$} - - %\tkzFct[domain=0:14,color=green,very thick]{15*\x**2 + 50*\x + 2000*0.81**\x} - \end{tikzpicture} - \end{minipage} - - \noindent - \textbf{Partie B} - - \noindent - Vos réponses aux questions suivantes ne pourront pas être justifiées à l'aide du graphique. - - \noindent - \begin{enumerate} - \item Calculer et interpréter $f(4)$ et $g(4)$. - \item Quelle sera la production de produit A 9 mois après le lancement? - \item À partir de la formule de $f(x)$ justifier que la fonction est décroissante. - \item Combien de temps faut-il attendre pour que la production de produit A soit inférieur à 400tonnes. - \item (*) Combien de temps faut-il attendre pour que la production totale soit supérieur à 2100 tonnes? - \end{enumerate} -\end{exercise} \collectexercisesstop{banque}