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Lynbrook West, an apartment complex, has 100 two-bedroom units. The monthly profit (in dollars) realized from renting out x apartments is given by the following function. p(x)=-12x^2+2160x-59000 To maximize the monthly rental profit, how many units should be rented out? units What is the maximum monthly profit realizable?

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Answer:

To maximize the monthly rental profit, 90 units should be rented out.

The maximum monthly profit realizable is $38,200.

Explanation:

Vertex of a quadratic function:

Suppose we have a quadratic function in the following format:


f(x) = ax^(2) + bx + c

It's vertex is the point
(x_(v), y_(v))

In which


x_(v) = -(b)/(2a)


y_(v) = -(\Delta)/(4a)

Where


\Delta = b^2-4ac

If a<0, the vertex is a maximum point, that is, the maximum value happens at
x_(v), and it's value is
y_(v).

In this question:

Quadratic equation with
a = -12, b = 2160, c = -59000

To maximize the monthly rental profit, how many units should be rented out?

This is the x-value of the vertex, so:


x_(v) = -(b)/(2a) = -(2160)/(2(-12)) = (2160)/(24) = 90

To maximize the monthly rental profit, 90 units should be rented out.

What is the maximum monthly profit realizable?

This is p(90). So


p(90) = -12(90)^2 + 2160(90) - 59000 = 38200

The maximum monthly profit realizable is $38,200.

User Ezaoutis
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