Problem 21 - Algebra and Trigonometry

Question

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. A theater is presenting a program for students and their parents on drinking and driving. The proceeds will be donated to a local alcohol information center. Admission is 2.00 for parents and 1.00 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

To obtain the exact solution one can follow the graphical method and see the effect of changing the ratio of the variables. However, one can also use calculus methods or even some programming languages with optimization packages to get more precise solutions.

Step 1: Set up the inequalities

First, let's denote the number of parents by \(p\) and the number of students by \(s\). The constraints we have are:\n\n1. The total number of people must not exceed 150:\(p + s \leq 150\)\n\n2. For every 2 parents there should be at least one student: \(s \geq \frac{p}{2}\)

Step 2: Graph the inequalities

First, graph the line \(p + s = 150\). This is a downward sloping line that crosses the \(p\) axis at 150 and the \(s\) axis at 150.\n\nThen, graph the line \(s = \frac{p}{2}\). This line crosses the \(p\) axis at 0 and the \(s\) axis at 0 and has a slope of 1/2. The feasible region will be the area on the graph where all the inequalities are satisfied.

Step 3: Identify the feasible region

The feasible region, which is the area satisfying all inequalities, will be the area that is below the line \(p + s = 150\) and above the line \(s = \frac{p}{2}\).

Step 4: Find the maximum of the function under the constraints

The function to maximize is the total proceeds from selling the tickets: \(F(p,s) = 2p + s\). Find the points at the vertices of the feasible region and evaluate the function at these points. The solution to the problem is the point that gives the maximum value of the function.

Key Concepts

Constraints

In linear programming, constraints are the conditions that a solution to the problem must satisfy. In the context of the theater program for students and parents, we have two main constraints. The first constraint is that the number of attendees cannot exceed the theater's capacity. This is expressed as: \( p + s \leq 150 \), where \( p \) is the number of parents, and \( s \) is the number of students. This constraint ensures that the number of people does not exceed 150, which is the theater's maximum capacity.
The second constraint deals with the relationship between the number of parents and students whereby every two parents must accompany at least one student, given by: \( s \geq \frac{p}{2} \). This constraint ensures that there are sufficient students in relation to the parents attending.

• These constraints are used to formulate the conditions that any permissible solution must adhere to.
• In a graph, constraints are typically represented by straight lines, and solutions must fall within the area defined by these lines.

Feasible Region

The feasible region is a vital concept in linear programming where it represents all the possible solutions that satisfy all the constraints. In terms of our theater case study, it’s the area where the conditions for maximum theater capacity and the ratio of students to parents are simultaneously satisfied.
To identify the feasible region:

• You graph the inequalities, which in this case are \( p + s \leq 150 \) and \( s \geq \frac{p}{2} \).
• The feasible region is visually the area where these lines overlap or intersect.

This region forms a polygon on the graph. Any solution within this polygon is a potential candidate for the optimal solution. The edges of this region are determined by the constraints, and the vertices are the key to finding the optimal solution.
In simpler terms, the feasible region is where all the mathematical conditions we are working with live and where we search for the best solution amongst many possible options.

Objective Function

In linear programming, the objective function is the expression that needs to be optimized, whether maximizing or minimizing the value. For our scenario involving theater ticket sales, the objective function is used to calculate the total income from the event.
Here, the objective function can be denoted as: \( F(p, s) = 2p + s \) This represents the total income from ticket sales, where \( p \) is the number of parents and \( s \) is the number of students. Each parent ticket sells for \(2, while a student ticket sells for \)1.

• The aim is to determine the number of parents and students that results in the highest possible revenue.
• After identifying the feasible region, this function is evaluated at each vertex of the polygon determined by the feasible region.

By evaluating the objective function at these vertices, we can find the combination of \( p \) and \( s \) that provides the maximum income from the ticket sales. This point is considered the optimal solution to the problem. The objective function essentially answers the question: "Given these constraints, how can we make the most money?"