[Math] Determine the maximum area of a norman window with a perimeter of $800$

calculus

I need help with the following question

Determine the dimensions of the window with the maximum area, if the perimeter of the window is $800$cm.

I have made a diagram of the window that was given to us. Please excuse my bad paint skills :)

enter image description here

I have done following work

$$ A_c = \frac{\pi r^2}{2} \\ A_r = lw \\ P = 2l + w + \pi w$$

Rearranging the perimeter equation for length

$$\begin{align}
&800 = 2l + w + \pi w \\
&\frac{800-2w-\pi w}{2} = l \\
&\frac{400-w-\pi w}{2} = l
\end{align}$$

Subbing $l$ into the Area equation

$$\begin{align}
&A(x) = lw + \frac{\pi w^2}{2} \\
&A(x) = \left(\frac{400-w-\pi w}{2}\right)(w) + \frac{\pi w^2}{2}\\
&A(x) = 400w-w^2\\
&A'(x) = 400-2w\\
&w = 200
\end{align}$$

The answer is supposed to be $86.2$ = $r$ and $h$ = $178.2$. Please point me the correct direction, I think I am making a really silly mistake with this one.

Thanks!

Best Answer

Between $\frac{800-2w-\pi w}{2} = l \\ \text { and } \frac{400-w-\pi w}{2} = l $

you didn't divide $\pi w$ or $l$ by $2$ when you divided the first two terms by $2$

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