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 30 feet?

Answer 1

let d = diameter and h = height
30 = (1/2)*d*Pi + d + 2h
h = (30 - (1/2)d*Pi-d)/2 = 15 - (1/4)d*Pi - d/2

Answer 2

area = (1/2)*Pi*(d/2)^2 + d * h

Answer 3

= (1/2)*Pi*(d/2)^2 + d(15 - (1/4)d*Pi - d/2) = (1/2)*Pi*(d/2)^2 + 15d - (1/4)d^2*Pi-(1/2)d^2

Answer 4

and at this point i get bored and use mathematica
Solve[D[(1/2)*Pi*(d/2)^2 + 15 d - (1/4) d^2*Pi - (1/2) d^2, {d,
1}] == 0, d]

Answer 5

{{d -> 60/(4 + \[Pi])}}

Answer 6

(1/2)*Pi*(d/2)^2 + 15 d - (1/4) d^2*Pi - (1/2) d^2 /.
d -> 60/(4 + \[Pi]) // N

Answer 7

63.0112

Answer 8

thanks!