Problem 16 - Precalculus: Functions and Graphs

Question

Machinery A small firm manufactures bookshelves and desks for microcomputers. For each product it is necessary to use a table saw and a power router. To manufacture each bookshelf, the saw must be used for \(\frac{1}{2}\) hour and the router for 1 hour. A desk requires the use of each machine for 2 hours. The profits are $$ 20\( per bookshelf and \)\$ 50$ per desk. If the saw can be used for 8 hours per day and the router for 12 hours per day, how many bookshelves and desks should be manufactured each day to maximize the profit?

Step-by-Step Solution

Answer

16 bookshelves and 0 desks should be manufactured each day to maximize profit.

Step 1: Define the Variables

Let's define the variables needed to solve the problem. Let \( x \) represent the number of bookshelves and \( y \) the number of desks. Our objective is to maximize the profit expressed in terms of \( x \) and \( y \).

Step 2: Write the Objective Function

The profit for each bookshelf is \\(20 and for each desk is \\)50. Therefore, the objective function to maximize the profit is \( P = 20x + 50y \).

Step 3: Set Up the Constraints

The constraints are given by the machine usage:1. For the table saw: \( \frac{1}{2}x + 2y \leq 8 \) (hours)2. For the power router: \( x + 2y \leq 12 \) (hours).Each constraint reflects the maximum capacity of each machine available per day.

Step 4: Graph the Constraints

Plot the inequalities on a graph. The area where the inequalities overlap represents feasible solutions. Determine the points of intersection by solving the equations \( \frac{1}{2}x + 2y = 8 \) and \( x + 2y = 12 \).

Step 5: Solve the System of Equations

To find the vertices of the feasible region, solve the system of equations:* From \( \frac{1}{2}x + 2y = 8 \): Multiply by 2 to simplify: \( x + 4y = 16 \)* Equation from \( x + 2y = 12 \).Subtract these equations:- \( x + 4y = 16 \)- \( -(x + 2y = 12) \)This gives: \( 2y = 4 \Rightarrow y = 2 \). Substituting \( y = 2 \) into \( x + 2y = 12 \), we get \( x + 4 = 12 \Rightarrow x = 8 \).

Step 6: Evaluate the Corner Points

The corner points of the feasible region are \((0, 0)\), \((0, 6)\), \((8, 2)\), and \((16, 0)\). Evaluate the profit function at each of these points:* At \((0, 0)\), \( P = 20 \times 0 + 50 \times 0 = 0 \).* At \((0, 6)\), \( P = 20 \times 0 + 50 \times 6 = 300 \).* At \((8, 2)\), \( P = 20 \times 8 + 50 \times 2 = 210 \).* At \((16, 0)\), \( P = 20 \times 16 + 50 \times 0 = 320 \).

Step 7: Identify the Solution

The maximum profit is at the point \((16, 0)\) which means the firm should manufacture 16 bookshelves and 0 desks each day.

Key Concepts

Objective Function

In linear programming, the objective function is the equation that you aim to maximize or minimize. It is the heart of the linear programming problem. Here, the objective function is crafted around profit maximization. For the given problem, the profit function is represented by \( P = 20x + 50y \). This means that for every bookshelf made, there's a profit of \\( 20\, and for each desk, \\) 50\.

• \( x \) is the number of bookshelves produced.
• \( y \) is the number of desks produced.

The goal is to discover the combination of \( x \) and \( y \) that gives the highest possible profit, while adhering to the resource constraints.

Constraints

Constraints in linear programming define the limitations or restrictions placed on the variables. They represent the hurdles that a company or a person must work within while trying to reach the optimal solution.
In this scenario, each bookshelf and desk requires a certain amount of time on a table saw and a power router. The constraints formed from this situation are:

• Time for table saw: \( \frac{1}{2}x + 2y \leq 8 \) hours
• Time for power router: \( x + 2y \leq 12 \) hours

These constraints ensure that the usage of each machine does not exceed available time per day. They guide the solution towards feasible and practical outcomes.

Feasible Region

The feasible region is the set of all possible points that satisfy the constraints. In linear programming, this region is often a polygonal area on a graph representing all the possible solutions to the problem under given constraints.
To define the feasible region, you must graph the constraints as inequalities and find the area where they overlap.
This graphically defined space will give you the possible combinations of bookshelves \( x \) and desks \( y \) that can be produced. In our case, solving the constraints provides critical interaction points like \( (0, 0) \), \( (0, 6) \), \( (8, 2) \), and \( (16, 0) \) defining the vertices of the feasible region.

Profit Maximization

Profit maximization is the end goal of this linear programming problem. After defining the objective function and constraints, the task is to evaluate the resulting feasible region, specifically at its corner points. Why? Because, due to the linear nature of the function and constraints, the optimal solution lies at one of these vertices.
Each corner represents a potential production combination. We calculate the profit at each point using:\[ P = 20x + 50y \]

• At \( (0, 0) \), the profit \( P = 0 \)
• At \( (0, 6) \), the profit \( P = 300 \)
• At \( (8, 2) \), the profit \( P = 210 \)
• At \( (16, 0) \), the profit \( P = 320 \)

The point \( (16, 0) \) offers the highest profit of \$ 320\, therefore, in order to maximize profits, 16 bookshelves should be produced per day.