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

A Norman window has the shape of a rectangle surmounted by a semicircle, as shown in the figure below. A Norman window with perimeter 30 ft is to be constructed. Find a function that models the area of the window. Find the dimensions of the window that admits the greatest amount of light. (Round your answers to one decimal place.)

Solution

Perimeter = 30 ft

radius of semicircle = x/2

x + 2height + pi*(x/2) = 30

x(1+0.5pi) + 2height = 30

height = 15 -x(0.5+ 0.25pi) = 15 - 1.28x

Area = x*height + (pi*r^2/2)/2

= x[15 -1.28x] + 0.785x^2

= 0.785x^2 +15x - 1.28x^2 = -0.495x^2 +15x

Maximum area occurs at x = -b/2a = -(15/(2*-0.785)

x = 9.6 ft

height = 15 -1.28*9.6 = 2.8 feet