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

Let x = the width of the rectangular part and y be the height. x is also the diameter of the semicircle. x/2

is the radius of the semicircle.

Perimeter, P = (semicircle perimeter)+(rectangle perimeter - top width)

= πx/2 + (2y+x) = 24

y = (24-x(1+π/2))/2 = 12-x(1+π/2)/2

Area = (Area of semicircle)+(Area of rectangle)

A=π(x/2)²/2 + xy

A(x) = πx²/8 + x(12-x(1+π/2)/2)

which can be simplified.