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This problem involves optimizing the production of two types of toy airplane models (A and B) based on constraints of assembly and packaging time, while maximizing the manufacturer's profit. Let's break it down step by step:

### Variables:
Let:
- $ x $ = number of units of **Model A** to be produced
- $ y $ = number of units of **Model B** to be produced

### Constraints:
- **Assembly time**:
  - Each unit of Model A takes 20 minutes to assemble.
  - Each unit of Model B takes 25 minutes to assemble.
  - The total available assembly time is 3000 minutes.
  \[
  20x + 25y \leq 3000
  \]

- **Packaging time**:
  - Each unit of Model A takes 10 minutes to package.
  - Each unit of Model B takes 5 minutes to package.
  - The total available packaging time is 1200 minutes.
  \[
  10x + 5y \leq 1200
  \]

### Objective:
- The goal is to **maximize profit**.
  - Profit for Model A: $9 per unit.
  - Profit for Model B: $12 per unit.
  The objective function to maximize is:
  \[
  Z = 9x + 12y
  \]

### Non-Negativity constraints:
\[
x \geq 0 \quad \text{and} \quad y \geq 0
\]

### Linear Programming Problem:
We now have the system of inequalities:
1. $ 20x + 25y \leq 3000 $
2. $ 10x + 5y \leq 1200 $
3. $ x \geq 0 $
4. $ y \geq 0 $

We need to solve this system graphically or using another method (e.g., simplex method) to find the values of $ x $ and $ y $ that maximize $ Z $.

Would you like further details on how to solve this step by step, or should I compute the solution directly?

Un fabricante produce dos modelos de aviones de juguete. El fabricante requiere 20 minutos para ensamblar el modelo A y 10 minutos para empaquetarlo. El fabricante requiere 25 minutos para ensamblar el modelo B y 5 minutos para empaquetarlo. En cierta semana, el tiempo total disponible para ensamblar es 3000 minutos, el tiempo total disponible para empaquetar es 1200 minutos. Cada avión del modelo A que se vende deja una ganancia de $9, y cada avión del modelo B que se vende deja una ganancia de $12. ¿Cuántas unidades de cada modelo debe producir para maximizar las ganancias de esta semana?

Learn how to solve a linear programming problem involving toy airplane production. Given constraints for assembly and packaging times, find the optimal number of Model A and Model B units to maximize profit. This step-by-step solution covers concepts in algebra and optimization, suitable for Grades 10-12.

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