Final answer:
The symphony should charge $37.50 per ticket to maximize the profit, determined by the vertex of the profit function, which is found using the vertex formula of a parabola.
Step-by-step explanation:
The question relates to finding the maximum profit which is a typical optimization problem in mathematics. The function provided, -20p^2 + 1500p, represents the revenue (and thus the profit since costs are fixed) the symphony earns per concert based on the ticket price p. To find the ticket price that maximizes profit, we need to determine the vertex of the parabola represented by this quadratic function.
The vertex formula of a parabola in standard form ax^2 + bx + c is given by -b/2a. In this case, our a is -20 and our b is 1500. Plugging these values into the vertex formula, we compute the ticket price that maximizes profit as p = -1500 / (2 * -20) = 37.5. Therefore, the symphony should charge $37.50 per ticket to achieve maximum profit.