# Norman Window?

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 12 feet?

Relevance
• A= xy + π*r²/2

12= x + 2y + π*r

r= (1/2)x

12= x + 2y + (1/2)π*x

y= 6 - x/2 -(1/4)π*x

A= x(6 -x/2 -π*x/4) + π/2(x/2)²

= 6x -x²/2 -π*x²/8

A'= 6 -x -π*x/4= 0

x(1 + π/4)= 6; x≈ 3.36 ft

y≈ 1.68 ft, r ≈ 1.68

maximum area= 3.36*1.68 + π*1.68²/2≈ 10.1 ft²

• P = 12

P = (pi/2) * d + d + 2x

12 = (pi/2) * d + d + 2x

24 = pi * d + d + 4x

24 = (pi + 1) * d + 4x

24 - (pi + 1) * d = 4x

(1/4) * (24 - (pi + 1) * d) = x

A = pi * (d/2)^2 * (1/2) + d * x

A = (pi/8) * d^2 + d * (1/4) * (24 - (pi + 1) * d)

A = (pi/8) * d^2 + 6d - (pi + 1) * d^2

A = (1/8) * (pi * d^2 + 48d - 8 * (pi + 1) * d^2)

A = (1/8) * (d^2 * (pi - 8pi - 8) + 48 * d)

A = (1/8) * (48 * d - (7pi + 8) * d^2)

dA/dd = (1/8) * (48 - 2 * (7pi + 8) * d)

dA/dd = 0

0 = (1/8) * 2 * (24 - (7pi + 8) * d)

0 = 24 - (7pi + 8) * d

(7pi + 8) * d = 24

d = 24 / (7 * pi + 8)

x = (1/4) * (24 - (pi + 1) * d)

x = (1/4) * (24 - (pi + 1) * 24 / (7pi + 8))

x = (24/4) * (1 - (pi + 1) / (7pi + 8))

x = 6 * (7pi + 8 - pi - 1) / (7pi + 8)

x = 6 * (6pi + 7) / (7pi + 8)

A = (1/8) * (48 * d - (7pi + 8) * d^2)

A = (1/8) * (48 * 24 / (7pi + 8)  -  (7pi + 8) * 24^2 / (7pi + 8)^2)

A = (1/8) * (48 * 24 / (7pi + 8) - 24 * 24 / (7pi + 8))

A = (1/8) * 24 * 24 * (2 - 1) / (7pi + 8)

A = 72 / (7pi + 8)

• Use surface area of sphere and divide by 2

A=( 4π6^2 )/2

• My window is much smaller than that.