Area of a Window

**lancelot854** · December 2nd 2010, 04:22 PM

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.

**skeeter** · December 2nd 2010, 04:28 PM

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

**lancelot854** · December 2nd 2010, 04:31 PM

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

**skeeter** · December 2nd 2010, 04:51 PM

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

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

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

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

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

$A$ will be a maximum when

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

**lancelot854** · December 2nd 2010, 05:08 PM

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

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