Two seperate questions. Please help explain steps. A Norman window has the shape

Question

Two seperate questions. Please help explain steps.

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See the figure below.) If the perimeter of the window is 24 ft, find the value of x so that the greatest possible amount of light is admitted. The manager of a 100 - unit apartment complex knows from experience that all units will be occupied if the rent is $500 per month. A market survey suggests that, on average, one additional unit will remain vacant for each $10 increase in rent. What rent should the manager charge to maximize revenue?

Explanation / Answer

All will be occupied when rent = 500 dollars

For every 10 dollar increase, one less unit will be occupied

Let x ---> units ocupied
y ---> rent

So, revenue, R = xy

R = (100 - x)(500 + 10x)

R = -10x^2 + 1000x - 500x + 50000

R = -10x^2 + 500x + 50000

dR/dx = -20x + 500 = 0

20x = 500

x = 25

So, when x = 25 ---> we have 100 - 25 --> 75 rooms occupied
And when x = 25, rent of the room --> 500 + 10x --> 500 + 10*25 --> $750

So, the manager must charge $750 per month ----> ANSWER

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Perimeter = 2x + 2y + pi*r

P = 2x + 2y + pi * x/2

24 = 2x + 2y + pi*x/2

48 = 4x + 4y + pi*x

y = 48 - (4 + pi)*x

Area = xy + (1/2)*pi*(x/2)^2

A = x(48 - 4x - pi*x) + (pi/8)*x^2

A = 48x - 4x^2 - pi*x^2 + (pi/8)x^2

A = (-7pi/8)x^2 - 4x^2 + 48x

dA/dx = (-7pi/8)*(2x) - 8x + 48 = 0

(-7pi/4)*x - 8x + 48 = 0

-7pi*x - 32x + 192 = 0

x = 192 / (7pi + 32) ----> ANSWER

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