Math Help - Area of a Window

1. Area of a Window

A Norman window is constucted by adjoining a semi-circle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet.

I know the equations:
$A=xy+\frac {pi*x^2}{4}$
$P=x+2y+\frac {pi*x}{2}=16$
(Don't know how to make the pi symbol in LaTeX, )

I tried to solve it multiple times but each time I end with something not possible, such as 4+1=0(just an example). Any help is appreciated, thanks. Here is a drawing of the figure, sorry for my bad paint skills.

2. Originally Posted by lancelot854
A Norman window is constucted by adjoining a semi-circle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet.

I know the equations:
$A=xy+\frac {pi*x^2}{4}$
$P=x+2y+\frac {pi*x}{2}=16$
(Don't know how to make the pi symbol in LaTeX, )

I tried to solve it multiple times but each time I end with something not possible, such as 4+1=0(just an example). Any help is appreciated, thanks. Here is a drawing of the figure, sorry for my bad paint skills.
area of the semicircle is $\displaystyle \frac{\pi \left(\frac{x}{2}\right)^2}{2} = \frac{\pi x^2}{8}$

3. Well, that's one mistake, but even so, I still get messed up along the way. -.-

4. $\displaystyle x\left(\frac{\pi + 2}{2}\right) + 2y = 16$

$\displaystyle y = 8 - x\left(\frac{\pi + 2}{4}\right)$

$\displaystyle A = 8x - x^2\left(\frac{\pi + 2}{4}\right) + \frac{\pi x^2}{8}$

$\displaystyle A = 8x - x^2 \left(\frac{\pi + 2}{4} -\frac{\pi}{8} \right)$

$\displaystyle A = 8x - x^2 \left(\frac{\pi + 4}{8}\right)$

$A$ will be a maximum when

$\displaystyle x = \frac{32}{\pi + 4}$

5. Could I ask how you came to find $\frac{\pi + 2}{2}$ and $\frac{\pi + 2}{4}$? Thanks for the help.

EDIT: Nevermind, figured it out after a lot of thinking. :P