https://r2.erweima.ai/mathbot/image/17261578924657dm3vnyb.jpg

This problem involves determining the number of units of each product (A, B, and C) that should be produced in order to use all the available machine hours optimally. The system can be represented as a system of linear equations. Here’s how we approach it:

### Step 1: Define the variables
Let:
- \( x_A \) = number of units of product A,
- \( x_B \) = number of units of product B,
- \( x_C \) = number of units of product C.

### Step 2: Matrix of machine times
From the given matrix, the time required to process each product on each machine is:

\[
\begin{bmatrix}
3 & 1 & 2 \\
1 & 2 & 4 \\
2 & 1 & 1
\end{bmatrix}
\]

The rows represent the machines I, II, and III, and the columns represent products A, B, and C, respectively. So, we can write the following system of equations based on the available machine hours:

1. \( 3x_A + x_B + 2x_C = 750 \) (Machine I)
2. \( x_A + 2x_B + 4x_C = 900 \) (Machine II)
3. \( 2x_A + x_B + x_C = 570 \) (Machine III)

### Step 3: Solve the system of equations
We need to solve this system to find the values of \( x_A \), \( x_B \), and \( x_C \).

I'll solve this system now.The solution to the system of equations is:

- \( x_A = 120 \) units of product A,
- \( x_B = 270 \) units of product B,
- \( x_C = 60 \) units of product C.

Thus, the correct answer is option **a**.

Would you like more details on how the system of equations was solved, or do you have any other questions?

### Related Questions:
1. How would the solution change if the total available hours for Machine I were increased?
2. What happens to the production if the processing time for product B on Machine II is doubled?
3. How can the system be solved graphically instead of algebraically?
4. How would additional constraints (e.g., production quotas) affect this problem?
5. Can this system be solved using a matrix inversion method?

**Tip:** When solving systems of equations involving real-world problems, it's useful to double-check that each solution makes sense physically or practically (e.g., non-negative values for units of production).

Una empresa produce tres productos, A, B y C, los que procesa en tres máquinas. El tiempo (en horas) requerido para procesar una unidad de cada producto por las tres máquinas está dado en una matriz.
Se dispone de la máquina I por 750 horas, de la máquina II por 900 horas y de la máquina III por 570 horas.
¿Cuántas unidades de cada producto deberían producirse con el objetivo de emplear todo el tiempo disponible de las máquinas?

Learn how to solve a matrix-based optimization problem to determine the number of units of three products processed by three machines with given machine hours. This exercise involves solving a system of linear equations, ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation