Question

The profit, P, (in dollars) for Ace Car Rental is given by P= 100x -0.17?, where x is the number of cars rented. How many cars have to be rented

for the company to maximize profits?

Answer 1

To maximize profits for Ace Car Rental using the equation P = 100x - 0.17x², the vertex of the quadratic equation needs to be found. The maximum profit occurs at the vertex, and the calculations lead to renting approximately 291 cars to achieve maximum profit.

Step-by-step explanation:

The profit P for Ace Car Rental is given by the quadratic equation P = 100x - 0.17x², where x is the number of cars rented. To find the number of cars that need to be rented to maximize profits, we need to complete the square or use the vertex form of a quadratic equation. Since the quadratic coefficient is negative (-0.17), the parabola opens downwards, implying that the vertex represents the maximum point.

First, let's convert the equation into vertex form, which is y = a(x-h)² + k, where (h, k) is the vertex of the parabola:

1. Convert the quadratic term: The coefficient of x² is -0.17, so we factor it out to set up completing the square.
2. Complete the square: (-0.17)(x² - (100/0.17)x). Then add and subtract the square of half the coefficient of x, which is (50/0.17)².
3. Rewrite the equation in vertex form: y = -0.17(x - 291.1765)² + P, where P is the maximum profit.

The vertex (h, k) occurs at x = h, so the number of cars that need to be rented to maximize profit is about 291 cars (rounded to the nearest whole number).