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

The answer is below

Step-by-step explanation:

Let x be the diameter of the semicircle. radius = x/2

The window is a combination of a rectangle and semicircle.

Width of window = diameter = x, let length of the window = y.

Perimeter of semicircle = πr = πx/2

Perimeter of window = x + y + y + πx/2

20 = x + 2y + πx/2

2y + x + πx/2 = 20

2y = 20 - x(1 - π/2)

y = 10 - x(1 - π/2)/2

Area of semicircle = (1/2)πr² = (1/2)π(x/2)²

Area of window = xy + (1/2)π(x/2)²

A = x(10 - x(1 - π/2)/2) + πx²/8

A = 10x - x² - πx²/4 + πx²/8

A = 10x - x² - πx²/8

The maximum area is at dA / dx = 0

dA / dx = 10 - 2x - 2πx/8

0 = 10 - 2x - πx / 4

2x + πx / 4 = 10

2.785x = 10

x = 3.59 feet

Maximum area = 10x - x² - πx²/8 = 10(3.59) - 3.59² - π(3.59²) / 8

Maximum area = 17.95 feet²

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