What is the area of the largest possible Norman window with a perimeter of 34 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 34 feet? 

Answer 1

I will set this up for you; you will need to work through the algebra. Draw a figure!

The width of the window is 2r and the height is h.

Perimeter = 34 = pi*r + 2r + 2h => h = (1/2)[34 - pi*r - 2r]

Area = (1/2)pi*r² + 2rh = (1/2)pi*r² + 2rh = (1/2)pi*r² + 2r{(1/2)(34 - pi*r - 2r)

You need to collect terms a

& set dA/dr = 0 to give you the value of r where the area is maximized.