A small furniture company is adding handmade chairs and tables to its product line. Each chair generates $20 profit, and each table generates $30 profit. Chairs require 3 hours of cutting time and 2 hours of assembly, while tables require 4 hours of cutting and 3 hours of assembly. If the cutting machine is available for 16 hours per week, and assembly is available for 18 hours per week, how many chairs and tables should be produced to maximize profit?

This problem can be formulated as a **linear programming** problem, where we maximize profit under the given constraints of cutting and assembly hours.

### Step 1: Define Variables
Let:
- $ x $ be the number of chairs produced per week.
- $ y $ be the number of tables produced per week.

### Step 2: Objective Function
The objective is to maximize profit, where:
- Profit per chair = $20
- Profit per table = $30

So, the **objective function** is:
\[
\text{Maximize} \: Z = 20x + 30y
\]
Where $ Z $ represents total profit.

### Step 3: Constraints
The constraints are based on the availability of cutting and assembly time.

#### Cutting time:
- Chairs require 3 hours of cutting.
- Tables require 4 hours of cutting.
- The total cutting time is limited to 16 hours per week.

Thus, the constraint is:
\[
3x + 4y \leq 16
\]

#### Assembly time:
- Chairs require 2 hours of assembly.
- Tables require 3 hours of assembly.
- The total assembly time is limited to 18 hours per week.

Thus, the constraint is:
\[
2x + 3y \leq 18
\]

### Step 4: Non-negativity Constraint
Since the number of chairs and tables cannot be negative:
\[
x \geq 0, \: y \geq 0
\]

### Step 5: Solve the Linear Programming Problem
We now have the following system to solve:

**Maximize:**
\[
Z = 20x + 30y
\]
**Subject to:**
\[
3x + 4y \leq 16
\]
\[
2x + 3y \leq 18
\]
\[
x \geq 0, \: y \geq 0
\]

We will solve this by identifying the corner points of the feasible region and evaluating the profit at each point.

### Step 6: Find the Corner Points
The corner points are where the constraints intersect or where they meet the axes. We will solve this step-by-step using substitution or elimination to find the points where the constraints intersect. I'll calculate them.The solution to the linear programming problem is:

- Produce **0 chairs**.
- Produce **4 tables**.

This will result in a maximum profit of \$120 per week.

Would you like more details on how this was calculated, or do you have any further questions?

Here are 5 related questions to consider:
1. What if the profit per chair was increased to \$25? How would the solution change?
2. How would the result be affected if the cutting machine time was increased to 20 hours per week?
3. What is the maximum number of tables that can be produced if no chairs are made?
4. What is the shadow price for the cutting and assembly time constraints?
5. Can you graphically represent the feasible region of this problem?

**Tip:** When dealing with linear programming problems, understanding the corner points of the feasible region is crucial, as the optimal solution always occurs at one of these points.

Learn how to maximize profit for a small furniture company by solving a linear programming problem involving handmade chairs and tables. Understand the constraints on cutting and assembly time, and apply optimization techniques to find the optimal production numbers. Suitable for Grades 10-12 and college students.

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