A furniture manufacturer produces chairs and sofas. Chairs require 20 feet of wood, 1 lb. of foam rubber, and 2 square yards of material. Sofas require 100 feet of wood, 50 lbs. of foam rubber, and 20 square yards of material. The manufacturer has 1900 feet of wood, 500 lbs. of foam rubber, and 240 square yards of material in stock. If chairs can be sold for $20 each and sofas for $300 each, how many of each should be produced to maximize income?

This problem is a typical **linear programming** problem where we are trying to maximize income given some resource constraints. Let's break this down step by step.

### Define Variables:
Let:
- $ x $ = number of chairs produced
- $ y $ = number of sofas produced

### Objective Function:
We want to **maximize** the income generated by selling chairs and sofas. The income from one chair is $20, and the income from one sofa is $300. Thus, the objective function is:

\[
\text{Maximize } Z = 20x + 300y
\]

### Constraints:
We have three resources (wood, foam rubber, and material) that limit production:

1. **Wood constraint**: Chairs use 20 feet of wood, and sofas use 100 feet of wood. The total available wood is 1900 feet:
   \[
   20x + 100y \leq 1900
   \]

2. **Foam rubber constraint**: Chairs use 1 lb of foam rubber, and sofas use 50 lbs of foam rubber. The total available foam rubber is 500 lbs:
   \[
   x + 50y \leq 500
   \]

3. **Material constraint**: Chairs use 2 square yards of material, and sofas use 20 square yards of material. The total available material is 240 square yards:
   \[
   2x + 20y \leq 240
   \]

Additionally, the number of chairs and sofas produced cannot be negative:
\[
x \geq 0 \quad \text{and} \quad y \geq 0
\]

### Step 1: Write the System of Inequalities
We now have the following system:

1. $ 20x + 100y \leq 1900 $ (wood)
2. $ x + 50y \leq 500 $ (foam rubber)
3. $ 2x + 20y \leq 240 $ (material)
4. $ x \geq 0 $ (non-negativity)
5. $ y \geq 0 $ (non-negativity)

### Step 2: Solve the System
To solve this linear programming problem, we will find the feasible region defined by these inequalities and then determine the vertex points. The maximum value of the objective function will occur at one of the vertices.

Let me calculate this for you.The optimal solution suggests producing:

- **25 chairs** and
- **9.5 sofas**.

However, producing half a sofa isn't practical, so the manufacturer will need to round the result to integers, potentially adjusting for feasibility in the context of constraints.

The **maximum income** based on this solution is **$3350**.

Would you like to explore rounding strategies or more details about the process? 

Here are 5 related questions for further learning:
1. How can rounding the number of sofas to an integer affect the solution?
2. What would happen to the optimal solution if the resources available changed?
3. How can we graphically interpret this problem using the feasible region?
4. How does linear programming apply to real-world manufacturing scenarios?
5. What alternative optimization methods can be used for non-linear problems?

**Tip:** In linear programming, the solution often lies at the vertices of the feasible region formed by the constraints.

This problem involves optimizing the production of chairs and sofas to maximize income given resource constraints. Learn how to apply linear programming techniques, including writing inequalities for constraints on wood, foam rubber, and material, and solving to find the maximum profit. Suitable for high school and college students.

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