https://file.aiquickdraw.com/mathbot/file/1728848652374xcsgtgkb.jpg

The problem involves creating a system of linear inequalities to represent the constraints in labor hours for a company manufacturing tables and chairs.

### Variables:
- Let \( x \) be the number of tables produced.
- Let \( y \) be the number of chairs produced.

### Labor Constraints:
1. **Assembly Labor**: 
   - Each table requires 9 labor-hours.
   - Each chair requires 8 labor-hours.
   - Maximum labor hours available for assembly per day = 300.

   The inequality for assembly is:
   \[
   9x + 8y \leq 300
   \]

2. **Finishing Labor**: 
   - Each table requires 5 labor-hours for finishing.
   - Each chair requires 2 labor-hours for finishing.
   - Maximum labor hours available for finishing per day = 115.

   The inequality for finishing is:
   \[
   5x + 2y \leq 115
   \]

### Non-negativity Constraints:
Since the company cannot produce a negative number of tables or chairs, the following inequalities must also be included:
\[
x \geq 0, \quad y \geq 0
\]

### Final System of Inequalities:
\[
\begin{aligned}
9x + 8y & \leq 300 \quad \text{(assembly constraint)} \\
5x + 2y & \leq 115 \quad \text{(finishing constraint)} \\
x & \geq 0 \quad \text{(non-negativity)} \\
y & \geq 0 \quad \text{(non-negativity)}
\end{aligned}
\]

This system represents the constraints based on the labor hours available for assembly and finishing of tables and chairs.

Would you like more details or have any further questions?

### 5 Relative Questions:
1. How do you graph a system of inequalities like this one?
2. What would the solution region look like for this system of inequalities?
3. How would the problem change if the maximum labor hours for assembly increased?
4. What is the impact of the assembly and finishing labor ratios on production limits?
5. How can you find the optimal number of tables and chairs to produce under these constraints?

### Tip:
When working with linear programming problems, always start by identifying constraints clearly and translating them into inequalities.

A furniture manufacturing company manufactures dining-room tables and chairs. A table requires 9 labor-hours for assembly and 5 labor-hours for finishing. A chair requires 8 labor-hours for assembly and 2 labor-hours for finishing. The maximum labor hours available per day for assembly and finishing are 300 and 115, respectively. Write a system of linear inequalities that represents this situation.

Learn how to solve a linear programming problem involving a furniture company that manufactures tables and chairs with specific labor constraints for assembly and finishing. The problem introduces the use of linear inequalities to model real-world production limits.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation