Marginal Revenue for an Apartment Complex

Lynbrook West, an apartment complex, has 100 two-bedroom units. The monthly profit (in dollars) realized from renting $x$ apartments is represented by the following function:

\[ P(x) = -11x^2 + 1750x - 60000 \]

(a) What is the actual profit realized from renting the 61st unit, assuming that 60 units have already been rented?

The actual profit from renting the 61st unit at Lynbrook West, assuming 60 units have been rented, is calculated by taking the difference of the profit function values at 61 and 60 units. The actual profit realized from renting the 61st unit is $2219.

Explanation:

To calculate the actual profit from renting the 61st unit, assuming 60 units have already been rented, we need to use the profit function P(x) = -11x2 + 1750x - 60000.

First, we find the profit for 60 units, which is P(60). Then we find the profit for 61 units, which is P(61). The actual profit from renting the 61st unit will be the difference between P(61) and P(60).

Calculate P(60):P(60) = -11(60)2 + 1750(60) - 60000 = -11(3600) + 105000 - 60000 = -39600 + 105000 - 60000 = 45600.
Calculate P(61):P(61) = -11(61)2 + 1750(61) - 60000 = -11(3721) + 106750 - 60000 = -40931 + 106750 - 60000 = 47819.
Find the difference:Actual profit from 61st unit = P(61) - P(60) = 47819 - 45600 = $2219.

The actual profit from renting the 61st unit is $2219.