\documentclass[12pt]{article}\usepackage{ammaths}\begin{document}




\partie{Homework \#11}
In the figure below, $[AB]$ is the diameter of the half-circle $\Co$ and its length is worth 8cm.$O$ is the midpoint of $[AB]$ and $C$ is the midpoint of the arc $\arc{AB}$.\\
Let $M$ a moving point onto the arc $\arc{AC}$. The parallel to $(AB)$ through $M$ cut $\Co$ in $N$ and $H$ is the projection of $M$ on $(AB)$.\\
The aim of this homework is to find the position of the point $M$ that maximize the perimeter of the trapezoid $AMNB$.
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\psline[linestyle=dashed,dash=2pt 2pt](2.63,3.01)(2.63,0)
\rput[bl](3.8,-0.34){$A$}
\rput[bl](-4.18,-0.34){$B$}
\rput[bl](0,4.1){$C$}
\rput[bl](-0.14,-0.34){$O$}
\rput[bl](2.72,3.14){$M$}
\rput[bl](-3,3.1){$N$}
\rput[bl](2.36,-0.34){$H$}
\end{pspicture*}$$
\vspace{-20pt}
\probpart{ : Geometric simulation}
\begin{enumerate}
 \item Use Geogebra to draw the figure.
 \item Move the point $M$ onto the arc $\arc{AC}$.\\
       Surmize the maximal value of the perimeter of $AMNB$.
\end{enumerate}

\probpart{ : Using a function}
\begin{enumerate}
\item Let $x$ represent the length of $AM$.
	\begin{enumerate}
	\item Compute the value of $AC$. Deduce the intervall of the values of $x$.
	\item Prove that $BN = x$.
	\end{enumerate}
\item  Let $AH = a$.\\
       Use Pythagorean theorem in two wise-chosen triangles, prove successively that :
	\begin{enumerate}
	 \item $MH^2=16-(4-a)^2$ ;
	 \item $MH^2=x^2-a^2$. 
	\end{enumerate}
\item Deduce from the previous questions that $a=\cfrac{x^2}{8}$ and then that $MN=8-\cfrac{x^2}{4}$.
\item Établir que le périmètre du trapèze $AMNB$ est donné en fonction de $x$ par l'expression $-\dfrac{x^2}{4}+2x+16$.
\item Let $f$ the function defined over $[0; 6]$ by : $f(x)=-\dfrac{x^2}{4}+2x+16$.
	\begin{enumerate}
	\item Graph, using Geogebra, the function $f$.
	\item According to the graph of $f$, draw the table of variation of the function.
	\item Surmize the position of $M$ for which the perimeter of $f$ is maximal.
	\end{enumerate}
\end{enumerate}


\end{document}
