Question

Owners of a car rental company that charges customers between $45 and $100 per day have determined that the number of cars rented per day n can be modeled by the linear function n(p)=1000-10p, where p is the daily rental charge. How much should the company charge each customer per day to maximize revenue?

Answer 1

To maximize revenue, the car rental company should charge $50 per customer per day, as determined by finding the vertex of the parabola representing the revenue function.

Step-by-step explanation:

The student has provided a linear function n(p) = 1000 - 10p, where n represents the number of cars rented per day and p is the daily rental charge. To maximize revenue, the company should find the price p that maximizes the product of p and n(p), which is the revenue R(p) = p × n(p). This occurs at the vertex of the parabola formed by the revenue function, which is a quadratic equation.

To find this vertex, we can use the fact that the vertex form of a parabola is R(p) = a(p - h)^2 + k, where (h, k) is the vertex of the parabola. Since R(p) will be a downward-facing parabola with a maximum point, the p-coordinate of the vertex will be the value that maximizes the revenue. The x-coordinate of the vertex of the parabola represented by the quadratic function is given by -b/(2a). In this case, a is -10 (from the linear function), and b is 1000 (the constant term). Therefore, the p-coordinate of the vertex is -1000/(2 × -10) = 50.

Therefore, to maximize revenue, the car rental company should charge $50 per customer per day.