Applied Optimization

Question

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 42 feet?

Answer 1

Let the rectangle be L by W , with r = W/2.

p = W + 2L + Wπ/2 = 42 ---> L = 21 - (1/2 + π/4)W

A = LW + π/8W² A = 21W - (1/2 + π/8)W²

A^' = 21 - (1 + π/4)W = 0

W = 84 / (4 + π). A_max = 1764 / (4 + π) - (1/2 + π/8)(84 / (4 + π))²