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The problem involves a linear programming (LP) model to determine the optimal number of units for two products, A and B, in order to maximize profit.

Here's a breakdown of the information provided:

### Given Data
- **Product A:**
  - Profit per unit: $7
  - Cutting time per unit: 4 hours
  - Finishing time per unit: 2 hours
- **Product B:**
  - Profit per unit: $5
  - Cutting time per unit: 3 hours
  - Finishing time per unit: 1 hour

### Resource Constraints
- **Cutting time available:** 240 hours
- **Finishing time available:** 100 hours

### Objective
Maximize profit by determining the optimal quantities of Product A ($x$) and Product B ($y$).

### Formulating the Linear Programming Model
1. **Objective Function** (to maximize):
   \[
   \text{Maximize } Z = 7x + 5y
   \]
   
2. **Constraints:**
   - **Cutting time:** $4x + 3y \leq 240$
   - **Finishing time:** $2x + y \leq 100$
   - **Non-negativity:** $x \geq 0$, $y \geq 0$

### Solution Method
To solve this LP model, use the **Simplex Method** or **Graphical Method** if desired. I can proceed with the steps of solving this, or set it up if you need further clarification.

Would you like the full solution for this setup?

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**Additional Questions to Explore:**
1. How would changing the profit per unit for Product A affect the solution?
2. What happens if the available cutting time is increased to 300 hours?
3. How would an additional constraint, such as limited storage, impact the model?
4. Could there be situations where producing only one product is optimal?
5. How does the solution change if the simplex method finds multiple optimal solutions?

**Tip:** Linear programming models help optimize resource allocation, especially in manufacturing and production planning contexts.

A company needs to decide how many units of Product A and Product B to produce to maximize profit, given constraints on resources. Product A provides $7 profit per unit, requires 4 hours for cutting, and 2 hours for finishing. Product B provides $5 profit per unit, requires 3 hours for cutting, and 1 hour for finishing. The available hours for cutting and finishing are 240 and 100, respectively.

Determine the optimal number of units for Product A and Product B to maximize profit using linear programming. This involves creating an objective function with constraints on available cutting and finishing hours, solved using the Simplex Method.

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