Problem 51 - Algebra and Trigonometry

Question

Manufacturing Furniture A furniture factory makes wooden tables, chairs, and armoires. Each piece of furniture requires three operations: cutting the wood, assembling, and finishing. Each operation requires the number of hours (h) given in the table. The workers in the factory can provide 300 hours of cutting, 400 hours of assembling, and 590 hours of finishing each work week. How many tables, chairs, and armoires should be produced so that all available labor-hours are used? Or is this impossible? $$\begin{array}{|c|ccc|}\hline & {\text { Table }} & {\text { Chair }} & {\text { Armoire }} \\ \hline \text { Cutting (h) } & {\frac{1}{2}} & {1} & {1} \\ {\text { Assembling (h) }} & {\frac{1}{2}} & {1 \frac{1}{2}} & {1} \\\ {\text { Finishing (h) }} & {1} & {1 \frac{1}{2}} & {2} \\ \hline \end{array}$$

Step-by-Step Solution

Answer

It is impossible to use all available labor-hours; no integer solution exists.

Step 1: Define Variables

Let \( x \) represent the number of tables, \( y \) the number of chairs, and \( z \) the number of armoires to be produced.

Step 2: Write Equations Based on Labor Hours

Using the given hours, form equations for each type of operation: - **Cutting:** \( \frac{1}{2}x + y + z = 300 \) - **Assembling:** \( \frac{1}{2}x + \frac{3}{2}y + z = 400 \) - **Finishing:** \( x + \frac{3}{2}y + 2z = 590 \)

Step 3: Eliminate Fractions in Equations

Multiply each term in the equations by 2 to eliminate fractions:- **Cutting:** \( x + 2y + 2z = 600 \) - **Assembling:** \( x + 3y + 2z = 800 \) - **Finishing:** \( 2x + 3y + 4z = 1180 \)

Step 4: Subtract First Equation from Second

Subtract the cutting equation from the assembling equation to eliminate \( x \): \((x + 3y + 2z) - (x + 2y + 2z) = 800 - 600\) which simplifies to \( y = 200 \).

Step 5: Substitute y=200 into Equations

Substitute \( y = 200 \) into the modified equations:- **Cutting equation:** \( x + 2(200) + 2z = 600 \) simplifies to \( x + 2z = 200 \).- **Finishing equation:** \( 2x + 3(200) + 4z = 1180 \) simplifies to \( 2x + 4z = 580 \).

Step 6: Solve System of Equations for x and z

Use substitution for \( x + 2z = 200 \) and \( 2x + 4z = 580 \): - Multiply the first equation by 2: \( 2x + 4z = 400 \)- Subtract this from the second equation: \((2x + 4z) - (2x + 4z) = 580 - 400\) - The result is a contradiction, indicating no solution exists.

Key Concepts

Systems of Linear Equations

A system of linear equations is simply a collection of several linear equations that share the same set of variables. In the context of our furniture manufacturing problem, the variables are the quantities of each type of furniture to be produced. Systems of linear equations are often used to solve problems where multiple conditions or equations need to be satisfied at the same time. Here, each equation represents a different operation: cutting, assembling, and finishing. These equations sum the labor hours used by tables, chairs, and armoires to the factory's constraints on available labor hours. For example, the cutting operation equation is: \( \frac{1}{2}x + y + z = 300 \) This shows that the sum of hours spent cutting for all three furniture items equals the 300 available hours. The goal in these problems is typically to find values for the variables that satisfy all the equations simultaneously, known as finding a solution to the system. If no such solution exists, as in this problem, it means that the current setup of conditions or constraints cannot be satisfied.

Labor-Hour Allocation

In manufacturing, labor-hour allocation refers to distributing available working hours among various tasks or products. This ensures each operation is efficiently planned based on the time required for completion. In our furniture example: - **Cutting requires:** \( \frac{1}{2} \) an hour for tables and 1 hour each for chairs and armoires.- **Assembling needs:** \( \frac{1}{2} \) an hour for tables, \( 1\frac{1}{2} \) hours for chairs, and 1 hour for armoires.- **Finishing takes:** 1 hour for tables, \( 1\frac{1}{2} \) hours for chairs, and 2 hours for armoires.The solution involves ensuring that the hours allocated do not exceed the available hours, which are 300, 400, and 590 for cutting, assembling, and finishing, respectively.Efficient labor-hour allocation helps a factory optimize its resources so that it can produce the maximum quantity of furniture possible without violating the available labor limits. If constraints can't be met, re-evaluating the setup or resource allocation is necessary, as was identified in our problem.

Optimization in Manufacturing

Optimization in manufacturing is the process of making production as efficient as possible by utilizing resources wisely. This means using labor, materials, and machines in a way that maximizes output or profit while minimizing waste and cost. For optimization to work successfully, it must take into account constraints, such as labor-hour limits or material availability. Specifically in linear programming, we identify these constraints through mathematical equations which demonstrate relationships between different variables and resources. In this problem, we aimed to determine how many tables, chairs, and armoires to produce without exceeding the given labor-hour constraints for cutting, assembling, and finishing. However, the solution reveals a contradiction, meaning that we need to revisit our conditions or constraints. Sometimes, the parameters or resources can be adjusted, like increasing labor hours or improving production efficiency, to make the mathematical model feasible for an optimal solution. Overall, effective optimization helps manufacturing operations run smoothly, ensuring that resources are allocated to meet production goals while maintaining high levels of efficiency.