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

the central idea is that:

individuals should refuse to cooperate with institutions they believe unjust.

thoreau felt  that governments were more often agents of corruption than justice, even democracies, and that just because the majority feels a certain way does not make it just or correct in absolute terms. he uses the fact that slavery exists in our democracy to support this idea.

he goes on to say that, as a result, people should live true to their own internal laws regarding right and wrong, as a product of reflection and a moral life, rather than following every law that society imposes. 

step-by-step explanation: