A firm produces three products P1 P2 and P3 processor on 3 machines M1 M2 and M3 before completion process 25 units of P1 or 50 units of P2 of 75 units of P3 per hour M2 and process 50 units of any one product for our M3 can process 50 25 and 100 units per hour of products P1 P2 and P3 respectively the processing arts available on machine M1 M2 and M3 are 12 and 13 respectively using Matrix negotiation find how many units of 3 can be produced per day

To solve this problem, we can represent the processing capacities of the machines using matrices and apply matrix multiplication techniques to determine the total number of units of products \( P_1 \), \( P_2 \), and \( P_3 \) that can be produced per day.

### Given:
- Machine \( M_1 \) can process:
  - 25 units of \( P_1 \) per hour,
  - 50 units of \( P_2 \) per hour,
  - 75 units of \( P_3 \) per hour.
- Machine \( M_2 \) can process:
  - 50 units of any one product per hour.
- Machine \( M_3 \) can process:
  - 50 units of \( P_1 \),
  - 25 units of \( P_2 \),
  - 100 units of \( P_3 \) per hour.

### Time available:
- \( M_1 \) is available for 12 hours per day.
- \( M_2 \) is available for 12 hours per day.
- \( M_3 \) is available for 13 hours per day.

### Matrix representation:
We can represent the processing rates for each machine in matrix form. Let \( M \) be the matrix of machine processing rates and \( H \) be the vector of available hours for each machine.

The processing rate matrix \( M \) will be:
\[
M = \begin{bmatrix}
25 & 50 & 75 \\
50 & 50 & 50 \\
50 & 25 & 100
\end{bmatrix}
\]
Where:
- Each row represents the units processed per hour for each machine (\( M_1 \), \( M_2 \), \( M_3 \)),
- Each column represents the products (\( P_1 \), \( P_2 \), \( P_3 \)).

The time available vector \( H \) is:
\[
H = \begin{bmatrix}
12 \\
12 \\
13
\end{bmatrix}
\]
Where each element represents the hours available per day for each machine.

### Finding the total units produced:
To find the total number of units produced for each product, we perform matrix multiplication between the processing rate matrix \( M \) and the available hours vector \( H \).

\[
\text{Total units per day} = M \times H
\]

Let's compute this matrix multiplication to determine how many units of each product can be produced per day.The total number of units produced per day for each product is as follows:

- \( P_1 \): 1,550 units per day,
- \( P_2 \): 1,525 units per day,
- \( P_3 \): 2,800 units per day.

Would you like more details on how the matrix multiplication was done, or do you have any questions?

Here are 5 related questions to consider:
1. How would the production change if the machines were available for more or fewer hours?
2. What happens if the machines have different processing rates for other products?
3. How would introducing a fourth product affect the matrix representation?
4. Can this matrix approach be used to optimize production across multiple factories?
5. How does the allocation of machine time impact the total number of units produced?

**Tip:** Matrix multiplication is a powerful tool for solving systems of equations in many real-world scenarios, especially when dealing with production and resource allocation.

A firm produces three products P1, P2, and P3 processed on 3 machines M1, M2, and M3 before completion. Machine M1 processes 25 units of P1, 50 units of P2, or 75 units of P3 per hour. M2 processes 50 units of any one product per hour. M3 can process 50 units of P1, 25 units of P2, or 100 units of P3 per hour. The processing hours available on machine M1, M2, and M3 are 12 and 13 hours respectively. Using matrix multiplication, find how many units of P1, P2, and P3 can be produced per day.

Learn how to use matrix multiplication to calculate the total daily production of products P1, P2, and P3 on machines M1, M2, and M3. Explore how to apply linear algebra to solve real-world production optimization problems. Suitable for high school students in Grades 10-12.

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