This problem is an optimization problem, specifically a linear programming (LP) problem. The goal is to maximize the company's profit subject to material and labor constraints. Let's define the variables and set up the problem.

Variables:

Let:

• $x_1$ = number of Mountain Bikes produced per week
• $x_2$ = number of Road Bikes produced per week

Objective Function:

The objective is to maximize profit, given by: $\text{Maximize } Z = 37x_1 + 45x_2$ Where:

• $37 is the profit per Mountain Bike ($x_1$)
• $45 is the profit per Road Bike ($x_2$)

Constraints:

The problem is subject to two main constraints: material and labor.

1. Material constraint: Each Mountain Bike requires 2 kg, and each Road Bike requires 3 kg of material. The company has a total of 327 kg of material: $2x_1 + 3x_2 \leq 327$

2. Labor constraint: Each Mountain Bike requires 3 hours, and each Road Bike requires 1 hour of labor. The company has a total of 291 hours of labor: $3x_1 + x_2 \leq 291$

Additionally, the number of bikes produced must be non-negative: $x_1 \geq 0 \quad \text{and} \quad x_2 \geq 0$

Solution Method:

To solve this LP problem, the most common methods are:

• Graphical method (suitable for two variables)
• Simplex method (used for larger problems or more variables)

Since there are only two variables, we can use the graphical method or solve it systematically. Would you prefer me to solve this manually or use a tool to compute the values?