Problem 17 - Thinking Mathematically

Question

A theater is presenting a program on drinking and driving for students and their parents. The proceeds will be donated to a local alcohol information center. Admission is \(\$ 2\) for parents and \(\$ 1\) for students. However, the situation has two constraints: The theater can hold no more than 150 people and every two parents must bring at least one student. How many parents and students should attend to raise the maximum amount of money?

Step-by-Step Solution

Answer

In the mentioned case, the maximum profit will be achieved when there are as many parents as possible, given the constraints. Detailed calculation will depend on the portrayal of this situation on a graph and identification of the corner points of the feasible region.

Step 1: Defining the variables

Let's denote the number of parents as \( P \) and the number of students as \( S \).

Step 2: Setting up the constraints

We can set up the following equations based on the problem: \(P + S \leq 150\) (the number of people should not exceed the theater's capacity) and \(S \geq 0.5P \) (for any two parents, there must be at least one student).

Step 3: Setting up the objective function

The theater earns \$2 per parent and \$1 per student. Hence, the earnings, or the sum that we need to maximize, is \( f(P,S) = 2P + S\).

Step 4: Solving the problem graphically

The feasible region determined by the constraints is a polygon. Identify this region on a graph and locate the corner points.

Step 5: Finding the maximum

Evaluate the function \(f(P, S) = 2P + S\), at each of the corner points. The largest of these values will be the maximum profit.

Key Concepts

Constraints

Constraints are the limitations or conditions that a problem must satisfy. In the theater problem, we have specific constraints that shape the solution. Think of constraints as rules of a game that define what is allowed and not allowed. Here, they consist of two main requirements:

• The total number of attendees cannot exceed 150, which ensures the theater is not overcrowded. This limit is represented by the equation \( P + S \leq 150 \).
• For at least every two parents, there must be one student present. This reflects a ratio constraint in the audience composition and is expressed as \( S \geq 0.5P \).

Constraints act as boundaries for your solution. They help define where you can "play" within a feasible region, which is crucial to find the optimal solution.

Objective Function

The objective function is what you want to achieve or optimize in a linear programming problem. It’s the goal of your exercise. In the theater situation, the objective function represents the total money raised and is formulated as: \( f(P, S) = 2P + S \).

This expression sums up all earnings from parents and students.

• Each parent contributes \(\\(2\)
• Each student contributes \(\\)1\)

The idea is to tweak the numbers of parents \(P\) and students \(S\) within the boundaries set by the constraints, aiming to achieve the highest total earnings. The objective function is the guiding star, showing you the direction to your most beneficial solution.

Feasible Region

The feasible region in a linear programming problem is the set of all possible solutions that satisfy the problem's constraints. Visually on a graph, it appears as a shaded area where all constraints overlap.

For our theater problem, graphing the constraints \( P + S \leq 150 \) and \( S \geq 0.5P \) reveals the feasible region:

• All points in this region are combinations of \(P\) and \(S\) that adhere to the limits of both attendance and ratio of parents to students.
• The region is typically a polygon, and the solution must lie within this space.

Identifying the feasible region helps refine possible answers, directing attention to only those solutions that are allowable under given constraints.

Optimization

Optimization in linear programming is about seeking the best possible solution within constraints. For the theater’s problem, this means finding the values of \(P\) (parents) and \(S\) (students) that maximize the total earnings.

The final step involves:

• Evaluating the objective function at each vertex, or corner, of the feasible region delineated by the constraints.
• Comparing these results to identify the highest value of \( f(P, S) = 2P + S \).

Through this method, you ensure that the chosen solution not only meets all constraints but also achieves the maximum profit possible. Simply put, optimization is about making the most out of what you have, perfectly balancing rules with desired outcomes.