Math Problem Statement

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.

Solution

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 xx be the number of tables produced.
  • Let yy 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≤3009x + 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≤1155x + 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≥0,y≥0x \geq 0, \quad y \geq 0

Final System of Inequalities:

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.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Systems of Inequalities
Algebra

Formulas

9x + 8y ≤ 300 (Assembly constraint)
5x + 2y ≤ 115 (Finishing constraint)
x ≥ 0, y ≥ 0 (Non-negativity constraints)

Theorems

Linear Inequality Theorems

Suitable Grade Level

Grades 10-12

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