Optimization Problems

Question

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 18 feet.

Answer 1

Let r be the radius of the semicircle, b be the base of the rectangle and h be the height of the rectangle.

Perimeter is half the circumference of a circle plus twice the height plus the base of the rectangle. The base of the rectangle is the diameter of the semicircle or 2r.

p = πr + 2h +2r = 18

h = (18-r(π+2))/2

The Area of the window is the area of the semicircle plus the area of the rectangle:

A = πr²/2 + 2rh

Substituting in for h:

A(r) = πr²/2 + (9 -r(π+2)/2)r

A'(r) = πr+9-r(π+2)

Setting A'(r) to zero and solving for r,

r = 4.5

h =6.43

base = 2r = 9