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. First, identify the function that represents the profit, which is given as \( P(n) = n(50 - 0.5n) - 100 \). This is a quadratic function in terms of \( n \).
2. To find the number of people that maximizes the profit, we need to find the critical points of the function. Start by taking the derivative of \( P(n) \) with respect to \( n \).
3. The derivative \( P'(n) \) can be found using the power rule and the constant rule. Calculate \( P'(n) = \frac{d}{dn}[n(50 - 0.5n) - 100] \).
4. Set the derivative \( P'(n) \) equal to zero to find the critical points: \( P'(n) = 0 \). Solve this equation for \( n \) to find the value that maximizes the profit.
5. After finding the critical point, verify that it is a maximum by using the second derivative test or by analyzing the sign changes of \( P'(n) \) around the critical point. Ensure that \( n \) is within the feasible range (0 to 100) for the problem.

Key Concepts

Profit Function

The profit function P(n) = n(50 - 0.5n) - 100 represents the relationship between the number of people n on a tour and the profit generated. It is a quadratic function, where the profit depends on the number of attendees, accounting for both revenue and costs. Understanding this function is crucial for determining how changes in n affect profit.

Maximization of Functions

Maximizing a function involves finding the input value that yields the highest output. In this context, we need to find the value of n that maximizes the profit function P(n). This typically involves taking the derivative of the function, setting it to zero, and solving for n to find critical points, which can indicate maximum profit.

Critical Points and Second Derivative Test

Critical points occur where the first derivative of a function is zero or undefined, indicating potential maxima or minima. To confirm whether a critical point is a maximum, the second derivative test can be applied. If the second derivative is negative at a critical point, it indicates a local maximum, which is essential for determining the optimal number of tour participants.