A Norman window is a rectangle with a semicircle on top Supp

A Norman window is a rectangle with a semicircle on top. Suppose that the perimeter of a particular Norman window is to be 28 feet. What should the rectangle\'s dimensions be in order to maximize the area of the window and, therefore, allow in as much light as possible? (Round your answers to two decimal places.)

Solution

Let length of window be l and width be w

now diameter of window = width of window

perimeter of window = 2l + w+pi*r= 2l + w +pi(w/2) = 28

2l + w(1 +pi/2) = 28

l = 14 - w(1 +pi/2)/2 = 14 - 1.27w

Area of window = l*w+ pi*r^2/2 = 14w - 1.27w^2+ pi(w/2)^2/2

= 14w - 1.27w^2 + 0.3926w^2

= 14w - 0.877w^2

Maximum area at vertex : w = -b/2a = - (14/2*-0.877) = 7.98 ft

find length l = 14 - 1.27*7.98 = 3.86 ft

Now find maximum area using = l*w = 30.828 = 30.83 ft^2