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
possibe Norman window with a perimeter of 36 feet?

Answer 1

hope this helps !! Area of rectangle = (2r)*h = 2rh
Area of semicircle = 1/2 pi r^2
So we want to maximize A = 2rh + 1/2 pi r^2
but there are two variables, so we'll first try to eliminate one of them.

Note that perimeter = 2h + 2r + pi r = 31
so h = 31 - r - pi/2 r = 31 - (1 + pi/2)r

Substitute that into our equation for area to get:
A = 2r(31-(1+pi/2)r) + 1/2 pi r^2
= 62r - 2r^2 - 1/2 pi r^2

Take the derivative:
dA/dr = 62 - 4r - pi r
and set it equal to zero:
0 = 62 - 4r - pi r
(4 + pi) r = 62
r = 62 / (4 + pi)

Now that you have the radius, you can find the area A by using the equation
A = 62r - 2r^2 - 1/2 pi r^2