A Norman window is a rectangle with a semicircle on top. Suppose that...?

Question

...Suppose that a particular Norman window is to be 24 ft. in perimeter. Also assume that the semicircular portion of the window is to be constructed of stained glass which transmits only half as much light as regular glass. What should the dimensions be in order to allow the maximum amount of light to enter through the window?

You don't have to work this whole problem out. I really just want to know instructions on HOW it should be done or approached (in terms of math). I'm mostly just unsure about the part that mentions the stained glass and from then on. Thanks!

Answer 1

Let
p = perimeter
r = radius semicircle
h = height rectangle
A = area
L = amount light

Given p = 24 ft., find the dimensions of the window to maximize the amount of light.

p = πr + (2h + 2r) = πr + 2h + 2r = 24
2h = 24 - 2r - πr
h = 12 - r - πr/2

A = Area semicircle + Area rectangle
A = πr²/2 + 2rh

L = (1/2)(πr²/2) + 2rh = πr²/4 + 2rh
L = πr²/4 + r(24 - 2r - πr) = πr²/4 + 24r - 2r² - πr²
L = 24r - (2 + 3π/4)r²

Take the derivative and set it equal to zero to find critical values.

dL/dr = 24 - (4 + 3π/2)r = 0
(4 + 3π/2)r = 24
r = 24 / (4 + 3π/2) = 48 / (8 + 3π)

h = 12 - r - πr/2 = 12 - (1 + π/2)r
h = 12 - (1 + π/2) [48/(8 + 3π)]
h = 12 - 24(2 + π)/(8 + 3π)
h = 12 - 24(π + 2)/(8 + 3π)

___________

As a check:

p = πr + 2h + 2r = 24
p = (π + 2)r + 2h = 24

p = (π + 2) [48/(8 + 3π)] + 2[12 - 24(π + 2)/(8 + 3π)]
p = 48(π + 2) / (8 + 3π) + 24 - 48(π + 2) / (8 + 3π)
p = 24

Answer 2

The stained glass should be minimized to allow for larger rectangular window which allows better light.
You need to compare area of the half circle with the rectangular area find the combination where the circles areas is twice as big the area it would have taken for the rectangular (or the area that could be spend on the rectangular part.