Answer:
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
