Help with a hard Calculus 1 class problem?

Question

A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted class that transmits only half as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Negect the thickness of the frame. PLEASE show work. Me, my classmates, and friends are really stumpted on this one!

Answer 1

Let
Ps = perimeter of the semicircle
Pr = perimeter of the rectangle
Pt = total perimeter of the window
r = radius of the semicircle
h = height of the rectangle
Ls = light transmittance of the semicircle
Lr = light transmittance of the rectangle
Lt = total light transmited through the window
As = area of the semicircle
Ar = area of the rectangle
PI = number pi

Let's start with the perimeter:
Perimiter of a semicircle has the formula
Ps = (PI)*r
For this rectangle the perimeter will be obtained by adding 3 sides, height + width + height, (in this case width is the same as 2 radii)
Pr = h + 2r + h = 2r + 2h
Total perimeter
Pt = Ps + Pr = (PI)*r + 2r + 2h = r(PI + 2) + 2h
Since the perimeter is fixed, we can express h in terms of r
Pt = r(PI + 2) + 2h
h = (Pt - (PI + 2)r) / 2

Now for the light transmited
semicircle transmits only half as much light as rectangle

Ls = (1/2)Lr
total light transmited
Lt = Ls*As + Lr*Ar
Lt = Ls*((PI*r^2)/2) + 2Ls*(h*2r)
Lt = Ls((PI*r^2)/2) + 4h*r)
And,
h = (Pt - (PI + 2)r) / 2
Thus,
Lt = Ls((PI*r^2)/2) + 4r*((Pt - (PI + 2)r) / 2))
Lt = Ls((PI*r^2)/2) + 2Pt*r - 2PI*r^2 - 4r^2)
Lt = Ls((-3PI*r^2)/2) + 2Pt*r - 4r^2)
Lt = Ls(2Pt*r - r^2((3/2)PI + 4))

to get the maximum d(Lt)/dr = 0
d(Lt)/dr = Ls(2Pt - 2r((3/2)PI + 4)) = 0
(2Pt - 2r((3/2)PI + 4)) = 0
r = 2Pt / (3PI + 8)
And,
h = (Pt - (PI + 2)r) / 2
h = Pt/2 - (2Pt / (3PI + 8))(PI + 2)/2
h = Pt(1/2 - (PI + 2)/(3PI + 8))
h = Pt(PI + 4)/(6PI + 16)

Therefore the proportions of the window are a semicricle with radius
r = 2Pt / (3PI + 8)
and a rectangle with a height
h = Pt(PI + 4)/(6PI + 16)