From the list below, choose the appropriate linear programming system for the given application problem.

A division of Finite Furniture Company manufactures dining room tables and chairs.  Each table requires 40 board feet of wood and 3 labor-hours.  Each chair requires 16 board feet of wood and 4 labor-hours.  The profit for each table is $45 and the profit for each chair is $20.  In the 16th week of the year, the company has 3200 board feet of wood available and 520 labor-hours available.  How many tables and chairs should Finite Furniture Company manufacture to maximize its profit?

From the list below, choose the appropriate linear programming system for the given application problem.

A division of Finite Furniture Company manufactures dining room tables and chairs.  Each table requires 40 board feet of wood and 3 labor-hours.  Each chair requires 16 board feet of wood and 4 labor-hours.  The profit for each table is $45 and the profit for each chair is $20.  In the 16th week of the year, the company has 3200 board feet of wood available and 520 labor-hours available.  How many tables and chairs should Finite Furniture Company manufacture to maximize its profit?

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To solve this problem using linear programming, you need to set up the objective function and the constraints based on the given information. Let's break it down:

### Step 1: Define the variables
- Let $ x $ represent the number of tables manufactured.
- Let $ y $ represent the number of chairs manufactured.

### Step 2: Set up the objective function
The objective function represents the total profit, which we want to maximize.
- The profit from each table is $45, and from each chair is $20.
- So, the objective function is:
\[
\text{Maximize } Z = 45x + 20y
\]

### Step 3: Set up the constraints
The constraints are based on the resources available (wood and labor hours).

1. **Wood constraint:**
   - Each table requires 40 board feet of wood.
   - Each chair requires 16 board feet of wood.
   - The total available wood is 3200 board feet.
   - So, the wood constraint is:
   \[
   40x + 16y \leq 3200
   \]

2. **Labor hours constraint:**
   - Each table requires 3 labor-hours.
   - Each chair requires 4 labor-hours.
   - The total available labor hours are 520.
   - So, the labor-hours constraint is:
   \[
   3x + 4y \leq 520
   \]

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

### Step 4: Linear Programming System
The full linear programming system is:

\[
\text{Maximize } Z = 45x + 20y
\]
Subject to:
\[
40x + 16y \leq 3200
\]
\[
3x + 4y \leq 520
\]
\[
x \geq 0, \quad y \geq 0
\]

This system is the appropriate one to model the problem for Finite Furniture Company.

Would you like to see more details on how to solve this system, or do you have any questions?

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**Relative Questions:**
1. What is the next step in solving this linear programming problem?
2. How do you graph the constraints to find the feasible region?
3. What is the significance of the feasible region in linear programming?
4. Can this linear programming problem have multiple optimal solutions?
5. How do you determine the optimal solution using the graphical method?
6. What are slack variables, and how are they used in linear programming?
7. How would the system change if the profit per table or chair increased?
8. What if additional resources (wood or labor hours) were available? How would that affect the system?

**Tip:** In linear programming, always ensure that your constraints accurately reflect the resources and limitations described in the problem.

Learn how to maximize profit using linear programming with a problem involving manufacturing dining room tables and chairs. Solve the objective function Z = 45x + 20y subject to constraints on wood and labor hours. Suitable for high school students and beyond.

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