A Norman window has the shape of a rectangle surmounted by a semicircle.

Question

If the perimeter of the window is 24 ft, express the area A of the window as a function of the width x of the window.

A(x) = ?

Answer 1

Draw and label a diagram!

The perimeter of the semi-circle is πx/2

The perimeter of the rectangle is 2h + x

24 = πx/2 + 2h + x

24 - πx/2 - x = 2h

12 - πx/4 - x/2 = h

Area of rectangle: 12x - πx²/4 - x²/ 2

Area of semi-cirlce: πx²/2

Area of window: 12x + πx²/4 - x²/2

Answer 2

Perimeter 24 = 2x + 2h, h being the height. This gives h = 12 - x. Area A(x) = x*h = x(12 - x) = 12x - x².