Question

How do you maximize a window that consists of an open rectangle topped by a semicircle and is to have a perimeter of 288 inches?

Answer 1

I found two values, for the width and height (of the rectangular part) of your window:
`80.6` in and `40.4` in

Explanation:

Considering your window as:

Perimeter is:
`P=2a+b+pi(b/2)=288`
so: `a=1/2[288-b-pi/2b]` (1)
Area is:
`A=b*a+pi/2(b/2)^2` using the value of `a` from (1):
`A=b1/2[288-b-pi/2b]+pi/2(b/2)^2=`
`=144b-b^2/2-pi/4b^2+pi/8b^2=144b-b^2(1/2+pi/4-pi/8)`
Maximize the area deriving it and setting it equal to zero:
`A'=144-2b(1/2+pi/4-pi/8)=0`
so that `b=80.6`in
So from (1): `a=40.4`in

Where I used for the semicircle:
Perimeter`=1/2*2pir`
Area`=1/2*pir^2`