calc
posted by Anonymous on .
Given that a window entails a rectangle capped by a semicircle, given that the semicircle’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 semicircle’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.

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 = 12xπx
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(12xπx) + (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)