What is the area of the largest possible Norman window with a perimeter of 43 feet?

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 43 feet? 

Answer 1

The perimeter of the Norman window can be found through the following formula:

43 = x + 2y + 2(x/2)pi/2

perimeter = bottom of the window + the 2 sides of the window + the circumference of the top part of the window/ 2 (because we only want the top half)

area of the window can be found through the following formula:

area = xy + (pi(x/2)^2)/2

area of the window = bottom length * side length + area of a circle/ 2 (because we only want the top half)

Solve the perimeter equation for y:

2y = 43 - x - (x/2)pi, y = (43 - x - (x/2)pi)/2

plug the equation into the area equation for y:

area = x(43 - x - (x/2)pi)/2 + (pi(x/2)^2)/2 = (43x - x^2 - (x^2/2)pi + pi(x/2)^2)/2 (multiply the x through and divide the whole equation by 2)

Take the derivative:

area' = (43 - 2x - x*pi + pi * x/2)/2 = (43 - 2x - pi*x/2)/2

set area' = 0 and solve for x:

0 = (43 - 2x - pi*x/2)/2, 0 = 43 - 2x - pi*x/2 (multiply both sides by 2), 2x + pi*x/2 = 43, x(2 + pi/2) = 43

x = 43/(2 + pi/2)

plug x (bottom length) back into y to find the side length:

y = (43 - 43/(2 + pi/2) - (43/(2 + pi/2)/2)pi)/2, y = (43 - 43/(2 + pi/2) - 43pi/(4 + pi))/2

now that we have x and y, plug them into the area equation and you have your answers (I am going to use decimals at this point for simplicity (they are approximate decimals rounded to 2 decimal places)) x=12.04, y=6.02

area = 12.04 * 6.02 + (pi(12.04/2)^2)/2 = 129.45

and we can confirm the x and y values are correct by plugging them into the perimeter formula to see if it matches:

perimeter = 43 = 12.04 + 2*6.02 + 2(12.04/2)pi/2 and it does = 43 so our answer is correct