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) = -10 x^2 + 1780 x - 54,000
To maximize the monthly rental profit, how many units should be rented out?
units
What is the maximum monthly profit realizable?
$
You know that for the parabola
y = ax^2 + bx + c
the vertex (maximum in this case) is at x = -b/2a
So, plug and chug.
To find the number of units that should be rented out in order to maximize the monthly rental profit, we need to find the x-value that corresponds to the vertex of the function.
The general form of a quadratic function is P(x) = ax^2 + bx + c, where a, b, and c are constants.
In this case, the function is P(x) = -10x^2 + 1780x - 54,000.
To find the x-coordinate of the vertex, we can use the formula x = -b / (2a). In this case, a = -10 and b = 1780.
Substituting these values into the formula, we get x = -1780 / (2*(-10)), which simplifies to x = 1780 / 20 = 89.
Therefore, to maximize the monthly rental profit, 89 units should be rented out.
To find the maximum monthly profit, we need to substitute this value back into the function.
P(89) = -10(89)^2 + 1780(89) - 54,000
Calculating this, we get P(89) = 79210 dollars.
Therefore, the maximum monthly profit realizable is $79,210.