Problem 12 - Precalculus: Functions and Graphs

Question

Maximizing gross profit A stationery company makes two types of notebooks: a deluxe notebook with subject dividers, which sells for $$ 4.00,\( and a regular notebook, which sells for 3.00 dollars . The production cost is 3.20 dollars for each deluxe notebook and $$ 2.60\) for each regular notebook. The company has the facilities to manufacture between 2000 and 3000 deluxe and between 3000 and 6000 regular notebooks, but not more than 7000 altogether. How many notebooks of each type should be manufactured to maximize the difference between the selling prices and the production costs?

Step-by-Step Solution

Answer

Manufacture 3000 deluxe and 4000 regular notebooks for maximum profit.

Step 1: Defining Variables

Let \( x \) represent the number of deluxe notebooks and \( y \) represent the number of regular notebooks. We need to determine the values of \( x \) and \( y \) within the given constraints to maximize profit.

Step 2: Formulating the Profit Function

The profit for each deluxe notebook is \( 4.00 - 3.20 = 0.80 \) dollars, and for each regular notebook, it is \( 3.00 - 2.60 = 0.40 \) dollars. The total profit function, \( P \), is given by \( P = 0.80x + 0.40y \). We want to maximize \( P \).

Step 3: Identifying the Constraints

We have the following constraints:1. \( 2000 \leq x \leq 3000 \)2. \( 3000 \leq y \leq 6000 \)3. \( x + y \leq 7000 \).

Step 4: Evaluating the Profit at Constraint Intersections

To maximize profit, evaluate \( P \) at the vertices of the feasible region formed by the constraints. The potential vertex points are: - \((x, y) = (2000, 3000)\)- \((x, y) = (2000, 5000)\)- \((x, y) = (3000, 3000)\)- \((x, y) = (3000, 4000)\).

Step 5: Calculating Profit at Each Vertex

Calculate \( P \) for each vertex:- At \((2000, 3000)\), \( P = 0.80(2000) + 0.40(3000) = 1600 + 1200 = 2800 \).- At \((2000, 5000)\), \( P = 0.80(2000) + 0.40(5000) = 1600 + 2000 = 3600 \).- At \((3000, 3000)\), \( P = 0.80(3000) + 0.40(3000) = 2400 + 1200 = 3600 \).- At \((3000, 4000)\), \( P = 0.80(3000) + 0.40(4000) = 2400 + 1600 = 4000 \).

Step 6: Selecting the Optimal Solution

The maximum profit, \( 4000 \) dollars, occurs at the point \((x, y) = (3000, 4000)\). Thus, the company should manufacture 3000 deluxe and 4000 regular notebooks to maximize profit.

Key Concepts

Profit Maximization

In the context of linear programming, profit maximization is a common objective where one aims to make the most financial gain from certain products or services. Here's how it works in this problem: the company produces two types of notebooks, each with its own selling price and cost.
The goal is to calculate the total profit by subtracting the total production cost from the total revenue.
In simpler terms, the profit from each type of notebook is:

• Deluxe Notebook Profit: \(4.00 (sell price) - \)3.20 (cost) = \(0.80
• Regular Notebook Profit: \)3.00 (sell price) - \(2.60 (cost) = \)0.40

The total profit function, in this case, is represented by the equation: \( P = 0.80x + 0.40y \), where \( x \) is the number of deluxe notebooks and \( y \) is the number of regular notebooks.
The objective is to find the values of \( x \) and \( y \) that will make \( P \) as large as possible, ensuring that the company earns the maximum profit possible under given constraints.

Constraints

Constraints in linear programming are the limitations or restrictions on the variables. They shape the problem and determine the feasible solutions.
In our notebook example, there are several key constraints to consider:

• Deluxe Notebooks: The company can make between 2,000 and 3,000 deluxe notebooks, stated as \( 2000 \leq x \leq 3000 \).
• Regular Notebooks: Production for regular notebooks ranges from 3,000 to 6,000, represented as \( 3000 \leq y \leq 6000 \).
• Total Production Capacity: The combined total of both types of notebooks should not exceed 7,000, expressed as \( x + y \leq 7000 \).

These constraints ensure that the production does not exceed physical or resource limits. They are crucial, as they help to pinpoint feasible combinations of \( x \) and \( y \) to achieve maximum profit.

Feasible Region

The feasible region in linear programming is where all constraints overlap. This area contains all the potential solutions that satisfy every constraint at once.
In graphical terms, it's the area on a graph where all the inequalities hold true.
Let's visualize it:

• The constraint \( 2000 \leq x \leq 3000 \) will form vertical lines on a graph.
• Similarly, \( 3000 \leq y \leq 6000 \) will form horizontal lines.
• The constraint \( x + y \leq 7000 \) will form a slanted line, closer to a diagonal.

Where all these lines intersect and form a shape is your feasible region. This region is vital because the optimal solution, especially in linear programming problems like this, often exists at the vertices or corners of this shape.

Vertex Evaluation

Vertex evaluation involves analyzing the corners, or vertices, of the feasible region to find the optimal solution.
Each vertex of the feasible region represents a potential solution, given it satisfies all constraints.
In this case, by evaluating profit \( P \) at each vertex, you identify where maximum profit occurs.

• Vertex \( (2000, 3000) \): Profit \( P = 0.80 \times 2000 + 0.40 \times 3000 = 2800 \).
• Vertex \( (2000, 5000) \): Profit \( P = 0.80 \times 2000 + 0.40 \times 5000 = 3600 \).
• Vertex \( (3000, 3000) \): Profit \( P = 0.80 \times 3000 + 0.40 \times 3000 = 3600 \).
• Vertex \( (3000, 4000) \): Profit \( P = 0.80 \times 3000 + 0.40 \times 4000 = 4000 \).

The vertex with the highest profit value is \( (3000, 4000) \), where the profit \( P \) is 4000 dollars.
Thus, producing 3000 deluxe and 4000 regular notebooks yields the highest profit.