A window is in the form of a rectangle capped by a semicircle. The width of the rectangular portion is equal to the diameter of the semicircle.

Question

A window is in the form of a rectangle capped by a semicircle. The width of the rectangular portion is equal to the diameter of the semicircle.

If the total perimeter of the window is 20 feet, what is the maximum possible area of the window? GIVE ME ONE OF THE FOLLOWING SOLUTIONS

40/pi+4

320/8+pi

200/4+pi  

  

  

Answer 1

Let x = diameter of circle = width of rectangle and y = length of rectangle.

Perimeter = π(x/2) + x + 2y = 20

2y = 20 - x - (π/2)x So, y = 10 - x/2 - (π/4)x

Area = A = xy + (1/2)π(x/2)² = x(10 - x/2 - (π/4)x) + πx² / 8

A = 10x - (1/2)x² - πx²/4 + πx²/8

A' = 10 - x - (π/2)x + (π/4)x = 0

x(1 + π/2 - π/4) = 10

x[(4 + π)/4) = 10

x = 40 / (4 + π)

Evaluate A for the value of x above to determine the maximum area.

Answer 2

https://www.youtube.com/watch?v=Eh6cJgbPNZ8

See the video.