https://file.aiquickdraw.com/mathbot/file/17307762194228v5uqmrq.jpg

To solve this problem, we need to maximize the profit by producing a combination of Model A and Model B patio chairs, considering the constraints given in the table. Here's a breakdown of the solution.

### Define Variables
Let:
- $ x $ = number of Model A chairs produced
- $ y $ = number of Model B chairs produced

### Objective Function
The objective is to maximize profit:
\[
\text{Profit} = 75x + 60y
\]

### Constraints
From the information provided, we have the following constraints:

1. **Carpentry Time**:  
   Each Model A chair requires 5 hours, and each Model B chair requires 4.5 hours.  
   Total carpentry hours cannot exceed 2800:
   \[
   5x + 4.5y \leq 2800
   \]

2. **Staining Time**:  
   Each Model A chair requires 1 hour, and each Model B chair requires 1 hour.  
   Total staining hours cannot exceed 800:
   \[
   x + y \leq 800
   \]

3. **Assembly Time**:  
   Each Model A chair requires 1.5 hours, and each Model B chair requires 1 hour.  
   Total assembly hours cannot exceed 600:
   \[
   1.5x + y \leq 600
   \]

4. **Contract Requirement for Model A**:  
   The company must produce at least 200 Model A chairs:
   \[
   x \geq 200
   \]

5. **Non-negativity Constraints**:
   \[
   x \geq 0, \quad y \geq 0
   \]

### Solution Approach
This is a linear programming problem. We can solve it by using the constraints and objective function with the simplex method or graphical analysis (if solving manually) or by using software like Excel Solver or other optimization tools to find the maximum profit.

### Solving
Using these constraints and maximizing $ 75x + 60y $, the solution would involve calculating the optimal values of $ x $ and $ y $ that satisfy all constraints.

**Let me know if you’d like a step-by-step guide for solving it using a specific tool (e.g., Excel Solver).**

---

### Related Questions

1. What would be the impact on maximum profit if the staining time available was increased to 1000 hours?
2. How would the solution change if the profit per Model B chair increased to $80?
3. What if the contract requirement for Model A chairs was removed? How would that affect the optimal solution?
4. How does changing the assembly time requirement for Model B chairs from 1 hour to 1.2 hours affect the outcome?
5. What would be the maximum profit if only Model A chairs were produced (ignoring Model B)?

---

### Tip
When dealing with optimization problems, always check if there are any binding constraints (ones that are fully used in the optimal solution) as they give insight into what limits the solution most.

In the picture above, what is the total maximum profit earned by making the optimal number of custom patio chairs?

Discover how to maximize profit in custom patio chair production by determining the optimal combination of Model A and Model B chairs. This problem involves linear programming with constraints in carpentry, staining, assembly, and contractual requirements. Learn how to apply optimization techniques for practical business solutions.

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