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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 144865458134381--
%%--Tranches d\'imposition--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
Un citoyen doit remplir sa d\'{e}claration d'imp\^{o}ts. Son revenu annuel est
$R=18\,000$ euros.

\begin{enumerate}
\item Il doit tout d'abord retrancher $10\%$ de son revenu annuel, puis de
nouveau $20\%$ sur la somme obtenue pr\'{e}c\'{e}demment.

\begin{enumerate}
\item Quel est le montant obtenu apr\`{e}s la r\'{e}duction de $10\%$?

\item V\'{e}rifier que le montant obtenu apr\`{e}s la seconde r\'{e}duction
est de $12\,960$ euros.

Cette somme not\'{e}e $R_{i}$ est le revenu imposable de ce citoyen.

\item Quel pourcentage $R_{i}$ repr\'{e}sente-t-il par rapport \`{a} $R$?
\end{enumerate}

\item Soit $I$ l'imp\^{o}t pay\'{e} par ce citoyen. Si $R_{i}\in
\lbrack8\,100;14\,300[$, alors $I=0,2\times R_{i}-1\,300$.

\begin{enumerate}
\item Calculer l'imp\^{o}t du citoyen.

\item Quel pourcentage $I$ repr\'{e}sente-t-il par rapport \`{a} $R$?
\end{enumerate}

\item Le citoyen se souvient que l'an pass\'{e} la formule pour calculer
l'imp\^{o}t \'{e}tait identique et qu'il avait alors pay\'{e}~$788$ euros. D\'{e}terminer:

\begin{enumerate}
\item son revenu imposable de l'an pass\'{e};

\item son salaire annuel de l'an pass\'{e}.
\end{enumerate}

\item \'{E}tude de la courbe des imp\^{o}ts $y$ (en euros) exprim\'{e}s en
fonction du revenu $x$ (en euros).

La courbe \ref{1604021430}, page \pageref{1604021430}, est une succession de
segments de droites qui correspondent ici \`{a} cinq tranches:
\[
\left\{  \mbox{
\begin{tabular}{*{3}{l}}
\textbullet{} tranche 1,& si $0\leqslant x\leqslant 4\,100$ &
alors $y=0$ \\
\textbullet{} tranche 2,& si $4\,100\leqslant x\leqslant 8\,100$ &
alors $y=ax+b$ \\         \textbullet{} tranche 3,& si $8\,100\leqslant x\leqslant 14\,300$ &
alors $y=0,2x-1\,300$ \\
\textbullet{} tranche 4,& si $14\,300\leqslant x\leqslant c$ &
alors $y=0,3x-2\,730$ \\
\textbullet{} tranche 5,& si $x\geqslant c$ &
alors $y=0,4x-5\,040$ \\
\end{tabular}}\right.
\]
D\'{e}terminer alg\'{e}briquement:

\begin{enumerate}
\item les r\'{e}els $a$ et $b$ qui donnent l'\'{e}quation pour la tranche 2.

\item la fronti\`{e}re $c$ entre les tranches 4 et 5 et l'ordonn\'{e}e du
point $M$.
\end{enumerate}
\end{enumerate}%

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\caption{$\quad$}%
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%% \psset{xunit=0.421,yunit=.78}
%% \psaxes[linewidth=0.6pt,Dx=5]{->}(0,0)(0,0)(39,11.666)
%% \uput{.2}[270](39,0){\small $\times 10^3$ euros}
%% \uput{.2}[0](0,11.666){\small $\times 10^3$ euros}
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%% {\footnotesize
%% \psline(4.1,0)(4.1,11)\uput{.5}[270](2.05,11){Tranche 1}
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%% \caption{\ }\label{1604021430}
%% \end{figure}

\end{exercise}


\end{document}