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

Question

3 years ago

9

et. 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?

1 answer:

tekilochka [14] · Answer 1 · 2022-05-31T15:07:40+03:00

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 = -30.85 + 61.74

A = 30.89 sq/ft is max area