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?

1 Answer
Jul 1, 2015

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:
enter image source here
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=80.6in
So from (1): a=40.4in

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