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+%%% End: + diff --git a/TST/DS/DS_21_06_02/exercises.tex b/TST/DS/DS_21_06_02/exercises.tex new file mode 100644 index 0000000..6a8e5ed --- /dev/null +++ b/TST/DS/DS_21_06_02/exercises.tex @@ -0,0 +1,127 @@ +\collectexercises{banque} + +\begin{exercise}[subtitle={Recyclage}, points=10, tribe={1}, type={Exercise}] + \begin{minipage}{0.5\linewidth} + Une entreprise est spécialisée dans le recyclage de bouteilles d'eau en plastique. + + Elle peut produire chaque jour entre $0$ et $10$ tonnes de plastique qu'elle revend en totalité au prix + unitaire de $700$~\euro{} la tonne. + + On rappelle que le coût moyen correspondant à la production de $x$ tonnes de plastique est défini par + $C_M(x) = \dfrac{C_T (x)}{x}$, où $C_T(x)$ est le coût total pour la production de $x$ tonnes de plastique. + + Le coût marginal, noté $C_m$, est le coût induit par la production d'une tonne de plastique supplémentaire + lorsqu'on a déjà produit $x$ tonnes de plastique. + + \smallskip + + Les parties A et B sont indépendantes. + + \medskip + + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[scale=0.3]{./fig/couts} + \end{minipage} + + \textbf{Partie A} + + \medskip + + Sur l'annexe sont tracées les courbes représentant les coûts moyen et marginal (en euro) en fonction + de la quantité de plastique produite (en tonne) ainsi que la droite représentant le prix de vente + unitaire. + + On admet que le coût moyen est minimal lorsqu'il est égal au coût marginal. + + \medskip + + \begin{enumerate} + \item Déterminer graphiquement la quantité de plastique que doit produire l'entreprise pour que le coût + moyen soit minimal. + \item Déterminer graphiquement ce coût moyen minimal et en déduire le coût total correspondant. + \end{enumerate} + + \bigskip + + \textbf{Partie B} + + \medskip + + On dit qu'il y a profit lorsque le prix de vente unitaire est strictement supérieur au coût moyen. + + On admet que le profit de l'entreprise est maximal lorsque le coût marginal est égal au prix de vente + unitaire. + + \medskip + + \begin{enumerate} + \item Pour quelles quantités de plastique produites, l'entreprise réalise-t-elle un profit ? Le résultat + sera donné sous la forme d'un intervalle. + \item Déterminer graphiquement la quantité de plastique que doit produire l'entreprise pour que le + profit soit maximal. + \item Quel est le coût moyen correspondant à cette production ? + \item En déduire le coût total correspondant. + \item Calculer le profit total maximal. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Chiffre d'affaires mondial}, points=8, tribe={1}, type={Exercise}] + Le tableau suivant donne le chiffre d'affaires mondial d'une entreprise entre 2010 et 2016 en millions d'euros. + + \begin{center} + \begin{tabularx}{\linewidth}{|m{3.cm}|*{7}{>{\centering \arraybackslash}X|}}\hline + Année &2010 &2011 &2012 &2013 &2014 &2015 &2016\\ \hline + Rang de l'année $x_i$ & 0 &1 &2 &3 &4 &5 &6\\ \hline + Chiffre d'affaires $y_i$ + (en millions d'euros) &18,3 &20,1 &23,3 &25,3 &27,8 &30,6 &32,4\\ \hline + \end{tabularx} + \end{center} + + \medskip + + \textbf{Partie A : étude d'un premier modèle} + + \medskip + + \begin{enumerate} + \item Sur le graphique donné à la fin de l'exercice , représenter le nuage de points de coordonnées $\left(x_i~;~y_i\right)$ pour $i$ variant de $0$ à $6$. + \item + \begin{enumerate} + \item À l'aide de la calculatrice, donner une équation de la droite d'ajustement affine de $y$ en $x$ obtenue par la méthode des moindres carrés. Les coefficients seront arrondis au centième. + + Dans la suite, on choisit la droite d d'équation $y = 2,4x + 18,1$ comme ajustement affine du nuage de points. + \item Tracer la droite $d$ sur le même graphique donné en annexe. + \end{enumerate} + \item En supposant que cet ajustement demeure valable pendant plusieurs années, donner par lecture graphique le chiffre d'affaires de cette entreprise en 2020. Arrondir au million près. + \end{enumerate} + + \medskip + + \textbf{Partie B : étude d'un second modèle} + + \medskip + + \begin{enumerate} + \setcounter{enumi}{4} + \item Déterminer, à l'aide du tableau, le taux d'évolution global du chiffre d'affaires de l'entreprise entre 2010 et 2016. On exprimera le résultat en pourcentage arrondi au centième. + \item Déterminer le taux d'évolution moyen annuel entre 2010 et 2016, exprimé en pourcentage arrondi à l'entier le plus proche. + \item On suppose que le taux d'évolution annuel sera de 10\,\% entre 2016 et 2020. Estimer le chiffre d'affaires de l'entreprise en 2020. Arrondir au million près. + \end{enumerate} + + \bigskip + + \begin{center} + \begin{tikzpicture}[baseline=(a.north), xscale=1, yscale=0.5] + \tkzInit[xmin=0,xmax=12,xstep=1, + ymin=0,ymax=52,ystep=2] + \tkzDrawX[label=Rang de l'année, above=10pt] + \tkzLabelX + \tkzDrawY[label=Chiffre d'affaire en millions d'euros, right=15pt] + \tkzLabelY + \tkzGrid + \end{tikzpicture} + \end{center} +\end{exercise} + +\collectexercisesstop{banque} diff --git a/TST/DS/DS_21_06_02/fig/couts.png b/TST/DS/DS_21_06_02/fig/couts.png new file mode 100644 index 0000000000000000000000000000000000000000..0049fec8f6e4f4ea916c8958a9a1b12594a02587 GIT binary patch literal 70244 zcmb@uXH-*L)HWKWh#*Bpq=^D5s1#A87e#C!O{7V0A|ky?C@M|5h=O!Q1qGx^3B4mC zqN375jWmIT9w3xEcYt%w`+ncJW85(=Kh819-Ya|W%r)11%3Q(fs#oY~*=bQI6#ccU ziZ@Uw3I!Bu=Q0g7ypweFnH&6%(pll!O&a*~qOo`kui0EK>AGmz-*#~~b+SZR+1cA# z3ObuRSz6jT-?4WgP*lmGP)AVL6ffNLNSy4w|BT~ToY?dgG$EmTJmU;CW^^>?^;`5aCaAHkeyf0(ko~jA0!X=or%LSKwX0KvZm&K 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