A furniture company has 400 board ft of teak wood and can sustain up to 450 hours of labor each week. Each chair produced requires 5 ft of wood and 10 hours of labor, and each table requires 20 ft of wood and 15 hours of labor. If a chair yields a profit of 45andatableyieldsaprofitof45*andatableyieldsaprofitof*80, what are the numbers of chairs and tables that should be produced each week in order to maximize the company’s profit?

A: 24 chairs and 14 tables
B: 20 chairs and 18 tables
C: 28 chairs and 10 tables
D: 14 chairs and 24 tables

To solve this problem, we'll need to formulate and solve a linear programming problem. Let's break it down step by step.

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

### Constraints:
1. **Wood constraint**: Each chair requires 5 ft of wood, and each table requires 20 ft of wood. The total available wood is 400 ft.
   \[
   5x + 20y \leq 400
   \]

2. **Labor constraint**: Each chair requires 10 hours of labor, and each table requires 15 hours of labor. The total available labor is 450 hours.
   \[
   10x + 15y \leq 450
   \]

3. **Non-negativity constraint**: The number of chairs and tables cannot be negative.
   \[
   x \geq 0, \quad y \geq 0
   \]

### Objective Function:
The profit from producing \( x \) chairs and \( y \) tables is given by:
\[
\text{Profit} = 45x + 80y
\]
We want to maximize this profit function under the given constraints.

### Solve the Problem:

1. **Wood constraint**:
   \[
   5x + 20y \leq 400 \quad \Rightarrow \quad x + 4y \leq 80
   \]

2. **Labor constraint**:
   \[
   10x + 15y \leq 450 \quad \Rightarrow \quad 2x + 3y \leq 90
   \]

### Check each of the options provided:
**Option A: 24 chairs and 14 tables**

- Wood: \( 5(24) + 20(14) = 120 + 280 = 400 \) (OK)
- Labor: \( 10(24) + 15(14) = 240 + 210 = 450 \) (OK)
- Profit: \( 45(24) + 80(14) = 1080 + 1120 = 2200 \)

**Option B: 20 chairs and 18 tables**

- Wood: \( 5(20) + 20(18) = 100 + 360 = 460 \) (Exceeds limit, NOT possible)

**Option C: 28 chairs and 10 tables**

- Wood: \( 5(28) + 20(10) = 140 + 200 = 340 \) (OK)
- Labor: \( 10(28) + 15(10) = 280 + 150 = 430 \) (OK)
- Profit: \( 45(28) + 80(10) = 1260 + 800 = 2060 \)

**Option D: 14 chairs and 24 tables**

- Wood: \( 5(14) + 20(24) = 70 + 480 = 550 \) (Exceeds limit, NOT possible)

### Conclusion:
The correct option that maximizes profit while satisfying all constraints is **Option A: 24 chairs and 14 tables** with a profit of **2200**.

Would you like to explore other scenarios or constraints?

### Related Questions:
1. How would the profit change if the wood or labor availability increased?
2. What happens to the optimal solution if the profit per chair or table changes?
3. How does adding a new type of product impact the production strategy?
4. What is the graphical method for solving this linear programming problem?
5. How do sensitivity analyses work in linear programming problems?

**Tip:** In linear programming, always check the feasibility of constraints before calculating the objective function to ensure the solution is valid.

Explore how to maximize profit for a furniture company using linear programming. Understand the constraints involving wood and labor, and optimize production of chairs and tables for maximum profit. Detailed solution provided with analysis of options.

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