Question

A Norman window is a rectangle with a semicircle on top. Suppose that the perimeter of a particular Norman window is to be 24 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?

Answer 1

Answer:

Step-by-step explanation:

Let a is the width of the window and diameter of the semicircle and let h be height of the rectangular portion of the window

:

Perimeter:

2h + a + .5x*pi = 21

2h + 2.57a = 21

2h = 21 - 2.57a

h = (10.5-1.285a)

:

What would be the window with the greatest area;

Area = semicircle + rectangle

Radius = .5a

A = (.5*pi*(.5a)^2) + h*a

Replace h with (10.5-1.285a

A = (1.57*.25a^2) + x(10.5-1.285a)

A = .3927a^2 - 1.285a^2 + 10.5a

A = -.8923a^2 + 10.5a

Find the max area by finding the axis of symmetry; x = -b/(2a)

a = 5.88 meter is the width with the greatest area

:

Find the max area

A = -.8923(5.88^2) + 10.5(5.88)

A = -.8923(5.88^2) + 10.5(5.88)

A = -30.85 + 61.74

A = 30.89 sq/ft is max area