Problem 40 - Algebra for College Students

Question

A manufacturer of small copiers makes a profit of \(\$ 200\) on a deluxe model and \(\$ 250\) on a standard model. The company wants to produce at least 50 deluxe models per week and at least 75 standard models per week. However, the weekly production is not to exceed 150 copiers. How many copiers of each kind should be produced in order to maximize the profit?

Step-by-Step Solution

Answer

Produce 50 deluxe and 100 standard models for maximum profit of $45000.

Step 1: Define the Variables

Let \( x \) represent the number of deluxe models and \( y \) represent the number of standard models produced per week.

Step 2: Formulate the Objective Function

The profit for each deluxe model is \( \\(200 \) and for each standard model is \( \\)250 \). Therefore, the total profit \( P \) can be expressed as:\[ P = 200x + 250y \] We aim to maximize this objective function.

Step 3: Establish the Constraints

Based on the problem, the production constraints are:1. At least 50 deluxe models: \( x \geq 50 \)2. At least 75 standard models: \( y \geq 75 \)3. Total production not exceeding 150 copiers: \( x + y \leq 150 \)

Step 4: Identify the Feasible Region

The feasible region is determined by the intersection of the constraints. Plot the inequalities on a graph, where the region that satisfies all constraints is the feasible region.

Step 5: Calculate the Vertices of the Feasible Region

Calculate the points of intersection of the inequalities:- Intersection of \( x = 50 \) and \( x + y = 150 \) gives (50, 100).- Intersection of \( y = 75 \) and \( x + y = 150 \) gives (75, 75).- Intersection of \( y = 75 \) and \( x = 50 \) gives (50, 75).

Step 6: Evaluate the Objective Function at Each Vertex

Calculate the profit at each vertex found in Step 5:- At (50, 100): \( P = 200(50) + 250(100) = 20000 + 25000 = 45000 \)- At (75, 75): \( P = 200(75) + 250(75) = 15000 + 18750 = 33750 \)- At (50, 75): \( P = 200(50) + 250(75) = 10000 + 18750 = 28750 \)

Step 7: Determine the Maximum Profit

Compare the calculated profits. The maximum profit is \( \$45000 \) at the point (50, 100).

Key Concepts

Objective Function

In the realm of linear programming, the **objective function** is the equation that one seeks to either maximize or minimize. It's a mathematical expression that defines the goal, often related to profit, cost, or some other measure of interest. For our given problem, the manufacturer wants to maximize profit earned from producing deluxe and standard copier models.

The objective function here is represented as:

• \[ P = 200x + 250y \]

where:

• \( x \) is the number of deluxe copiers produced, with each adding \( \\(200 \) to the profit.
• \( y \) is the number of standard copiers produced, contributing \( \\)250 \) each.

To determine the best mix of copiers to produce, we have to maximize this function. Our task is to find the values of \( x \) and \( y \) that would make the profit \( P \) as large as possible, while remaining within defined boundaries.

Feasible Region

The **feasible region** is the set of all possible solutions that satisfy the constraints of a linear programming problem. To visualize it, you can plot the inequalities on a graph. The area where all these constraints overlap is the feasible region. It defines the permissible values for \( x \) and \( y \), within which the objective function should be optimized.

In our copier production problem, the constraints form a bounded region on the graph:

• The region must adhere to at least 50 deluxe models, specified by the line \( x = 50 \).
• At least 75 standard models should be produced, creating the boundary \( y = 75 \).
• The total cannot exceed 150 copiers, defining the line \( x + y = 150 \).

These boundaries intersect to form the feasible region. This region is crucial as any potential solution that can maximize the objective function must lie within it.

Constraints

**Constraints** are the conditions that solutions to a linear programming problem must satisfy. They limit the possible values of the variables in question.

• First, the manufacturer must produce at least 50 deluxe models, which gives the constraint \( x \geq 50 \).
• Similarly, for standard models, at least 75 are required, leading to the constraint \( y \geq 75 \).
• The total output must not exceed 150 copiers, so \( x + y \leq 150 \) represents this limit.

By establishing these constraints, the problem narrows the possible solutions to those which fit within these specified borders. This ensures any calculated solution is viable within real-world production limits.

Vertices Calculation

In the process of linear programming, calculating the **vertices** of the feasible region is critical because the maximum or minimum value of the objective function will occur at one of these vertices. You calculate where the constraints intersect, as these points form the corners of the feasible region.

For our problem:

• The intersection of \( x = 50 \) and \( x + y = 150 \) gives (50, 100).
• The intersection of \( y = 75 \) and \( x + y = 150 \) yields (75, 75).
• The corner at (50, 75) comes from the intersection of \( y = 75 \) and \( x = 50 \).

By evaluating the objective function at each of these vertices, one can determine where the function acquires its optimal value, in this case, maximizing the profit.