A Norman window is constructed by adjoining a semicircular window to the top of a rectangular window (See Figure 13.).

Question

A Norman window is constructed by adjoining a semicircular window to the top of a rectangular window (See Figure 13.). If the perimeter of the window in 16 ft, find its dimensions so that it admits the greatest amount of light possible.

Answer 1

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