A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus, the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 25 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.

Question

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus, the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 25 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.
What is the total height?

Steve · Answer

if the rectangular window has width 2x and height h, then the area of the whole shape is

a = 2xh + pi/2 x^2

You also know that x+2h+pi*x = 25, so solve for h and plug into the formula for area. The set da/dx = 0.