Math 110 Course Resources - Optimization Course Packet on applications: Maximizing profit by
The Correct Answer and Explanation is:
Here are the correct answers and a detailed explanation for the problem.
(a) Mt. Nittany Property Management
Number of apartments = 62
Maximum monthly profit = 10,220 dollars
(b) Happy Valley Rentals
Number of apartments = 50
Maximum monthly profits = 9,500 dollars
Explanation
This problem requires finding the number of apartments, x, that maximizes the monthly profit function, P(x) = -5x² + 620x – 9000. This function is a quadratic equation, and its graph is a downward opening parabola because the coefficient of the x² term (-5) is negative. The highest point of this parabola, called the vertex, represents the maximum possible profit.
The x-coordinate of the vertex gives the number of units (apartments) needed to achieve maximum profit. We can find this using the vertex formula x = -b / (2a). For this profit function, a = -5 and b = 620.
x = -620 / (2 * -5)
x = -620 / -10
x = 62
This calculation shows that renting 62 apartments will yield the absolute maximum profit.
For part (a), Mt. Nittany Property Management manages 100 apartments. Since the ideal number of 62 apartments is within their capacity (62 is less than 100), they can rent out exactly 62 apartments to maximize their profit. To find this maximum profit, we substitute x = 62 back into the profit function:
P(62) = -5(62)² + 620(62) – 9000
P(62) = -5(3844) + 38440 – 9000
P(62) = -19220 + 38440 – 9000
P(62) = 10,220
So, their maximum monthly profit is $10,220.
For part (b), Happy Valley Rentals only manages 50 apartments. They cannot rent out 62 apartments because they do not have that many. The vertex at x = 62 is outside their possible rental range of 0 to 50. Since the profit function increases up to the vertex, the highest profit for this company will occur at the maximum number of apartments they can rent, which is 50. We calculate the profit for x = 50:
P(50) = -5(50)² + 620(50) – 9000
P(50) = -5(2500) + 31000 – 9000
P(50) = -12500 + 31000 – 9000
P(50) = 9,500
Therefore, their maximum monthly profit is $9,500.
