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) = -8x2 + 1760x - 31000
(a) What is the actual profit realized from renting the 61st unit, assuming that 60 units have already been rented?
$ ??
(b) Compute the marginal profit when x = 60 and compare your results with that obtained in part (a).
$ ??
46592
To find the profit realized from renting the 61st unit, we can substitute x = 61 into the profit function P(x) and calculate the result.
(a) Let's substitute x = 61 into the profit function:
P(61) = -8(61)^2 + 1760(61) - 31000
Now, we can simplify the expression and calculate the result:
P(61) = -8(3721) + 107360 - 31000
P(61) = -29768 + 107360 - 31000
P(61) = 46592
Therefore, the actual profit realized from renting the 61st unit is $46,592.
(b) To compute the marginal profit when x = 60, we need to find the derivative of the profit function P(x) with respect to x. The derivative represents the rate of change of profit with respect to the number of apartments rented.
First, let's find the derivative of P(x):
P'(x) = -16x + 1760
Now, substitute x = 60 into the derivative function:
P'(60) = -16(60) + 1760
P'(60) = -960 + 1760
P'(60) = 800
The marginal profit when x = 60 is $800.
Comparing the results from part (a) and (b), the actual profit realized from renting the 61st unit is $46,592, whereas the marginal profit when x = 60 is $800. These values represent different aspects of profit: the actual profit reflects the total money earned from renting a specific number of units, while the marginal profit represents the additional profit gained from renting one more unit.