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


\en
\partie{Homework \#4}

In the figure below, $ABC$ is an isosceles triangle with both lengths $AB$ and $AC$ equal to 10 cm. The
length $BC$ is variable, let's denote it $x$, in cm. The point $H$ is the foot of the altitude passing through $A$.\\
The aim of this homework is to find the length of $BC$ that maximizes the area of the triangle $ABC$.
%The figure
\psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
$$\begin{pspicture*}(-0.6,-0.62)(4.46,4.5)
\pspolygon(2.42,0)(2.42,0.42)(2,0.42)(2,0)
\psline(2,4)(0,0)
\psline(2,4)(4,0)
\psline(2,4)(2,0)
\psline(0,0)(4,0)
\rput[bl](2.08,4.12){$A$}
\rput[bl](-0.6,-0.3){$B$}
\rput[bl](0.4,2.24){$10$}
\rput[bl](4.2,-0.3){$C$}
\rput[bl](3.1,2.24){$10$}
\rput[bl](1.8,-0.34){$H$}
%\rput[bl](2.38,0.56){$90°$}
\end{pspicture*}$$
% \columnbreak
\probpart{ -- Using Geogebra}
\begin{enumerate}
\item Use Geogebra to draw the figure. Keep in mind that the triangle must be isosceles with two sides equal to 10cm.
Find out how to display the area of the triangle on the picture. When you're done, send your file by internet to your
teacher.
\item Change the length of $BC$. Conjecture the maximal value of the area of $ABC$.
\end{enumerate}

\probpart{ -- Using a function}

\begin{enumerate}
\item Compute the area of the triangle when $x = 5$.
\item Is it possible to have $x=30$ ? What is the maximal value for $x$ ?
\item What is the minimal value for $x$ ? Explain quickly your answer.
\item Write the length of the altitude $AH$ as a function of $x$. 
\item Let $f$ be the function that maps any adequate value of $x$ to the corresponding area of $ABC$. Prove that
$f(x)=\dfrac{x}{4}\sqrt{400-x^2}$.
\item Draw the graph of $f$.
\item Let $\alpha$ be the value for which $f$ is maximal.  
\begin{enumerate}    
\item Using your calculator, build the table of values for $f$ between 1 and 20, with step $1$, and give two consecutive
values bounding $\alpha$.
\item Compute a new table of values for $f$ between your previous answers, with step $0.1$. Deduce a better
interval for $\alpha$.
\item In the same way, give an interval with diameter $0.001$ including $\alpha$.
\end{enumerate}
\end{enumerate}
% \end{multicols}


\end{document}
