Maximizing profit Suppose a tour guide has a bus that holds a maximum of 100 people. Assume his profit (in dollars) for taking people on a city tour is P(n) = n(50 - 0.5n) - 100. (Although P is defined only for positive integers, treat it as a continuous function.)


a. How many people should the guide take on a tour to maximize the profit?

Verified step by step guidance

1. Identify the profit function given by P(n) = n(50 - 0.5n) - 100, where n represents the number of people on the tour.
2. Rewrite the profit function in standard quadratic form, which is P(n) = -0.5n^2 + 50n - 100.
3. Determine the vertex of the quadratic function, as the maximum profit occurs at the vertex. The n-coordinate of the vertex can be found using the formula n = -b/(2a), where a and b are the coefficients from the standard form.
4. Substitute the values of a and b from the profit function into the vertex formula to find the optimal number of people, n.
5. Since the problem states that n should be a positive integer, round the result to the nearest whole number if necessary, ensuring it does not exceed the bus capacity of 100.