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We need to construct a mathematical model that describes the plant operating at full capacity. The departments involved are Cutting, Assembly, and Packaging, each with their labor-hour limits. We are given the labor requirements for each boat type (One-Person, Two-Person, and Four-Person). The goal is to set up equations for each department and solve them using Gauss-Jordan elimination.

Let:
- \( x_1 \) represent the number of one-person boats.
- \( x_2 \) represent the number of two-person boats.
- \( x_3 \) represent the number of four-person boats.

### 1. Cutting Department Equation:
The cutting department has a total of 425 hours available. From the table:
- One-Person Boat requires 0.5 hours of cutting.
- Two-Person Boat requires 1.0 hours of cutting.
- Four-Person Boat requires 1.5 hours of cutting.

Thus, the equation for the cutting department is:
\[
0.5x_1 + 1.0x_2 + 1.5x_3 = 425
\]

### 2. Assembly Department Equation:
The assembly department has a total of 472 hours available. From the table:
- One-Person Boat requires 0.8 hours of assembly.
- Two-Person Boat requires 1.2 hours of assembly.
- Four-Person Boat requires 1.6 hours of assembly.

Thus, the equation for the assembly department is:
\[
0.8x_1 + 1.2x_2 + 1.6x_3 = 472
\]

### 3. Packaging Department Equation:
The packaging department has a total of 157 hours available. From the table:
- One-Person Boat requires 0.1 hours of packaging.
- Two-Person Boat requires 0.3 hours of packaging.
- Four-Person Boat requires 0.6 hours of packaging.



Construct a mathematical model that describes the plant operating at full capacity. The departments involved are Cutting, Assembly, and Packaging, each with their labor-hour limits. Solve the system using Gauss-Jordan elimination.

Learn how to solve a system of equations using Gauss-Jordan elimination in this manufacturing plant optimization problem. This problem covers key linear algebra concepts, including forming equations for cutting, assembly, and packaging labor in boat production. Suitable for advanced high school and undergraduate students.

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