Question

A window has the shape of a rectangle surmounted by a semicircle similar to older buildings. The diameter of the semicircle is equal to the width of the rectangle. If the perimeter of the window is 30ft, find the dimensions of the window so that the greatest possible amount of light is admitted.

Answer 1

Solution: Let the dimensions of the rectangular part of the window be a x b. Since the diameter of the semicircle is equal to the width of the rectangle, therefore:

Given the perimeter of the window to be 30 feet. The perimeter of the window includes 2 length sides, on width side, and the perimeter P of the circle, therefore,

... (1)

For the amount of the light to be maximum, the area of the window should be maximum. The area of the window is given by:

Substituting the value of a from the equation (1), we'll get:

... (2)

To maximize the area A, differentiating the equation above w.r.t b and equating it to zero to find its critical point, we'll get:

Since we get only one critical point, therefore, there is no need to use the second derivative test to verify our answer. This value of b will give the maximum value of area. Substituting this value of b in the equation (1), we'll get the following value of y:

Therefore, the required dimension is 4.2 ft x 8.4 ft.

I hope it helps you!