Problem 51 - Precalculus: Mathematical for Calculus

Question

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? $$\begin{array}{|l|ccc|} \hline & \text { Table } & \text { Chair } & \text { Armoire } \\ \hline \text { Cutting (h) } & \frac{1}{2} & 1 & 1 \\ \text { Assembling (h) } & \frac{1}{2} & 1 \frac{1}{2} & 1 \\ \text { Finishing (h) } & 1 & 1 \frac{1}{2} & 2 \end{array}$$

Step-by-Step Solution

Answer

It is impossible to exactly match all available labor-hours within the production constraints.

Step 1: Define Variables

Let \( x \) be the number of tables, \( y \) be the number of chairs, and \( z \) be the number of armoires produced. We need to determine \( x \), \( y \), and \( z \) such that all the available labor-hours are utilized.

Step 2: Set Up the System of Equations

Using the given hours for each operation, form the following equations based on the available hours:1. Cutting: \( \frac{1}{2}x + y + z = 300 \)2. Assembling: \( \frac{1}{2}x + \frac{3}{2}y + z = 400 \)3. Finishing: \( x + \frac{3}{2}y + 2z = 590 \)

Step 3: Multiply Equations to Eliminate Fractions

To eliminate fractions, multiply the first two equations by 2:1. Cutting: \( x + 2y + 2z = 600 \)2. Assembling: \( x + 3y + 2z = 800 \)The third equation remains:3. Finishing: \( x + \frac{3}{2}y + 2z = 590 \)

Step 4: Simplify the System

Subtract the Cutting equation from the Assembling equation:\((x + 3y + 2z) - (x + 2y + 2z) = 800 - 600 \)This simplifies to:\( y = 200 \).

Step 5: Substitute to Find Another Variable

Substitute \( y = 200 \) into the Cutting equation: \( x + 2(200) + 2z = 600 \) \( x + 400 + 2z = 600 \)\( x + 2z = 200 \).

Step 6: Solve Third Equation with Found y

Substitute \( y = 200 \) into the Finishing equation: \( x + \frac{3}{2}(200) + 2z = 590 \)\( x + 300 + 2z = 590 \)\( x + 2z = 290 \).

Step 7: Solve for z Using Consistent Equations

Subtract \( x + 2z = 200 \) from \( x + 2z = 290 \):\( (x + 2z) - (x + 2z) = 290 - 200 \)\( 0 = 90 \), which yields an inconsistency, indicating a contradiction.

Step 8: Conclusion

The equations provide an inconsistency (90 ≠ 0) implying it's impossible to use all available labor-hours exactly. Therefore, the production goals cannot be met within the constraints without exceeding the available labor.

Key Concepts

System of Equations

In problems like the furniture factory's, setting up a system of equations is crucial to understand how to allocate resources effectively. Each piece of furniture requires a certain amount of labor in different operations like cutting, assembling, and finishing.
To determine how many tables, chairs, and armoires can be produced, we use equations to represent the total labor-hours available for each operation.

• For cutting, the equation is: \( \frac{1}{2}x + y + z = 300 \)
• For assembling, it becomes: \( \frac{1}{2}x + \frac{3}{2}y + z = 400 \)
• For finishing, it is: \( x + \frac{3}{2}y + 2z = 590 \)

These equations collectively form a system of linear equations that needs to be solved together. They represent the constraints imposed by the labor-hours in the problem.

Labor-Hours Optimization

Optimizing labor-hours means using available resources in the most efficient way possible. Here, the challenge is to produce as many tables, chairs, and armoires without exceeding the given labor constraints.
In this exercise, after setting up the initial equations, we modified them to remove fractions for easier calculations. This led to a refined system:

• Cutting: \( x + 2y + 2z = 600 \)
• Assembling: \( x + 3y + 2z = 800 \)
• Finishing: \( x + \frac{3}{2}y + 2z = 590 \)

However, the process revealed inconsistencies. An ideal optimization would show a clear solution, but here, the simplifications led to a logical contradiction \( 0 = 90 \), indicating the given resources cannot satisfy the constraints for all operations as initially hoped.

Resource Allocation

Resource allocation is the practice of distributing available resources to meet production targets effectively. In the context of the furniture factory, this involves determining the optimal number of tables, chairs, and armoires to produce, given the labor-hour constraints.
The key is to balance the requirements of each type of furniture against the available labor hours for cutting, assembling, and finishing.
The step-by-step solution attempted to allocate the labor-hours by calculating the feasible number of each furniture item using the solved equations. Unfortunately, it concluded with a realization that the target might be unachievable without exceeding available labor for at least one of the operations.
This outcome suggests that a reevaluation of the resource allocation might be necessary to better fit the labor availability or to possibly change production targets.