Expert: Ahmed Salami - 4/14/2010

Question
A church window consists of a blue semicircular section surmounting a clear rectangular section. The blue glass lets through half as much light per unit area as the clear glass. Find the radius of the window that admits the most light if the perimeter of the window is P feet.

Answer
Hi Paul,
The amount of light through any section is kA, where A is the area of that section and k is some constant factor.
If the amount of light through the blue semicircular section is k(πr²/2), then the amount of light through the clear rectangular section would be (2k)(2r.l) where l is the length of the rectangular section.
The total amount of light through the window is therefore
L = kπr²/2 + 4krl
But, with a perimeter of P
P = πr + 2l + 2r
 = r(π + 2) + 2l
l = [P - r(π + 2)]/2
and then
L = kπr²/2 + 4kr[P - r(π + 2)]/2
 = (k/2)[πr² + 4rP - 4r²(π + 2)]
Differentiating with respect to r,
dL/dr = 2πr + 4P - 8r(π + 2)
     = 2πr + 4P - 8πr - 16r
     = -r(6π + 16) + 4P
The maximum value of L occurs when dL/dr = 0 i.e
-r(6π + 16) + 4P = 0
r(6π + 16) = 4P
r = 4P/(6π + 16)

Regards