Question

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). The perimeter of the window is 20 feet.

A window is composed of a semicircle attached by its flat side to the top of a rectangle. The vertical side of the rectangle is labeled y, and the horizontal side of the rectangle is labeled x.
(a)
Write the area A of the window as a function of x.
A(x) =

(b)
What dimensions (in ft) will produce a window of maximum area? (Round your answers to two decimal places.)
width x =
ft
length y =
ft

Answer 1

16

Explanation: left and right sides are about 2.24 feet each
bottom = 4.48 feet
radius of the top semi-circle = about 2.24 feet
those dimensions maximizes the window area, which maximizes light

Perimeter = 16 = 2L+2r + pir = 2(2.24) + 2(2.24) + 3.14(2.24)
= 7.14(2.24) = 16

L=length of the right side = length of the left side
2L = 16-2r- pir

Area = 2Lr + pir^2/2
= (16-2r -pir)r + pir^2/2
= 16r -2r^2 -pir^2 +pir^2/2
=16r -2r^2 -pir^2/2

A' = 16 -4r -pir = 0
r = 16/(4+pi) = 16/7.14
r= 2.24 feet
L= (16-2(2.24)-pi(2.24)/2
= 2.24 feet

Perimeter = 2L+2r + pir
= 2(2.24) + 2(2.24) + pi(2.24)
=(2+2+pi)(2.24)
= 7.14(2.24)
= 16