Question

Given that a window entails a rectangle capped by a semi-circle, given that the semi-circle’s diameter concides with the rectangle’s width, given that the window’s outside (linear and curvilinear) perimeter is 24 feet, and given that the semi-circle’s stained glass transmits half the light of the rectangle’s unstained glass, determine the window’s rectangular and circular dimensions that will maximize the light transmitted.

Answer 1

Let the width of the rectangle be 2x , making the radius of the semicircle = x
let the height of the rectangle be y
then 2x + 2y + 2πx = 24
x + y + πx = 12
y = 12-x-πx

Assume that the amount of light (L) is a function of the area
L= area of rectangle + (1/2) area of semicircle
= 2xy + (1/2)(1/2)π x^2
= 2x(12-x-πx) + (1/4)π x^2 = 24x - 2x^2 - 2πx^2 + (1/4)π x^2
dL/dx = 24 - 4x - 4πx + (1/2)πx
= 0 for a max of L

times 2
48 - 8x - 8πx + πx = 0
48 = x(8 + 8π - π)
x = 48/(8+7π) = appr1.6
then r = 5.37

State the conclusion
(check my arithmetic, I should have written it out on paper)

Answer 2

A D C C C