Question

The profit from selling local ballet tickets depends on the ticket price. Using past receipts, we find that the profit can be modeled by the function Px=-15x²+600x+60, where P is the profit (in dollars) and x is the price of each ticket (in dollars). Find the ticket price that gives the maximum profit, and also find that maximum profit. Interpretation: What type of function is P(x)?

Answer 1

The ticket price that gives the maximum profit is $20, and the maximum profit is $2310. The function P(x) is a quadratic function that represents a downward-opening parabola.

Step-by-step explanation:

The profit function is given by P(x) = -15x²+600x+60, where P is the profit in dollars and x is the price of each ticket in dollars. To find the ticket price that gives the maximum profit, we need to find the vertex of the parabola represented by the profit function. The vertex of a parabola is given by the formula x = -/2, where a and b are the coefficients of the quadratic equation. In this case, a = -15 and b = 600. Plugging these values into the formula, we get x = -600/(2*-15) = 20. Therefore, the ticket price that gives the maximum profit is $20. To find the maximum profit, we substitute x = 20 into the profit function: P(20) = -15(20)²+600(20)+60 = $2310. Therefore, the maximum profit is $2310.

The function P(x) is a quadratic function, which represents a parabola. The negative coefficient of the x² term (-15) indicates that the parabola opens downwards. The maximum point of the parabola represents the highest profit, as in this case.