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The question asks about optimizing production to maximize profit, and this can be framed as a **linear programming problem**. Let's break down the setup.

### Problem Summary:
- **T50 DVD Player**:  
  - Uses 4 ounces of plastic  
  - Uses 4 ounces of metal  
  - Profit: \$10 per unit  
  - Max units: 20 units per week

- **G150 DVD Player**:  
  - Uses 2 ounces of plastic  
  - Uses 8 ounces of metal  
  - Profit: \$18 per unit  

- **Available materials**:  
  - 164 ounces of plastic  
  - 464 ounces of metal  

### Goal:
Maximize **profit** by determining how many units of **T50** and **G150** should be produced, subject to material and production constraints.

---

### Variables:
- Let $x$ = number of **T50 units**  
- Let $y$ = number of **G150 units**

---

### Constraints:
1. **Plastic Constraint**:  
   $4x + 2y \leq 164$  (Total plastic used)

2. **Metal Constraint**:  
   $4x + 8y \leq 464$  (Total metal used)

3. **Production Limit for T50**:  
   $x \leq 20$  

4. **Non-negativity**:  
   $x \geq 0$, $y \geq 0$

---

### Objective Function:
Maximize **profit**:  
\[
\text{Profit} = 10x + 18y
\]

---

### Solution Approach:
We will solve this using **linear programming** to find the values of $x$ and $y$ that maximize the profit while satisfying the constraints.

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Would you like me to solve this system step-by-step or perform the calculations directly to give you the optimal solution?

Guadalupe runs a factory that makes DVD players. Each T50 takes 4 ounces of plastic and 4 ounces of metal. Each G150 requires 2 ounces of plastic and 8 ounces of metal. The factory has 164 ounces of plastic, 464 ounces of metal available, with a maximum of 20 T50 that can be built each week. If each T50 generates $10 in profit, and each G150 generates $18, how many of each of the DVD players should Guadalupe have the factory make each week to make the most profit?

This problem involves optimizing production in a factory that makes two types of DVD players (T50 and G150) to maximize profit. The solution involves setting up linear constraints based on material usage and finding the optimal number of each product that should be manufactured for maximum profit. This linear programming problem is suitable for students in Grades 11-12 and college-level courses.

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