A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 47 feet?

Answer 1

p = W + 2L + πW/2 = 47
2L = 47 - W - πW/2
L = 23.5 - 1.28W

A = WL + π(W/2)^2/2
= W(23.5 - 1.28W) + .327W^2
= 23.5W - 1.28W^2 + .327W^2
= 23.5W - .953W^2

max area at W = 23.5/1.906 = 12.3 ft
L = 7.756

perimeter = 12.3 + 2*7.756 + 1.57*12.3 = 47.1 (close enough)

area = 12.3*7.756 + 3.14*6.15 = 114.7 ft^2

Answer 2

it says incorrect