A company produces two types of bicycles; mountain bikes and racing bikes. It takes 5 hours of assembly time and 2 hours of mechanical tuning to produce a mountain bike. It takes 8 hours of assembly time and 3 hours of mechanical tuning to produce a racing bike. The company has at most 24 hours of mechanical tuning labor per week and at most 171 hours of assembly labor per week. The company's profit is USD 120 for each mountain bike produced and USD 100 for each racing bike produced. The company wants to make as much money as possible. Let  x = the number of mountain bikes they produce, and let  y = the number of racing bikes they produce. What are the constraints for this problem?
 5x + 2y ≤ 171,   8x + 3y ≤ 24,   x ≥ 0,   y ≥ 0


 2x + 8y ≤ 171,   5x + 3y ≤ 24,   x ≥ 0,   y ≥ 0


 5x + 8y ≤ 171,   2x + 3y ≥ 24,   x ≥ 0,   y ≥ 0


 5x + 8y ≥ 171,   2x + 3y ≤ 24,   x ≥ 0,   y ≥ 0


 5x + 8y ≤ 171,   2x + 3y ≤ 24,   x ≥ 0,   y ≥ 0


The constraints in this problem are based on the available labor and time limits for both the assembly and mechanical tuning tasks. Here's how they break down:

1. **Assembly time constraint**: 
   - Each mountain bike requires 5 hours of assembly time, and each racing bike requires 8 hours of assembly time.
   - The company has at most 171 hours of assembly labor per week.
   - This gives the inequality:  
   \( 5x + 8y \leq 171 \)

2. **Mechanical tuning constraint**: 
   - Each mountain bike requires 2 hours of mechanical tuning, and each racing bike requires 3 hours of mechanical tuning.
   - The company has at most 24 hours of mechanical tuning labor per week.
   - This gives the inequality:  
   \( 2x + 3y \leq 24 \)

3. **Non-negativity constraints**: 
   - The number of mountain bikes and racing bikes produced cannot be negative, so:
   - \( x \geq 0 \)
   - \( y \geq 0 \)

Thus, the correct constraints are:

\[
5x + 8y \leq 171, \quad 2x + 3y \leq 24, \quad x \geq 0, \quad y \geq 0
\]

This matches option **5**:
\[
5x + 8y \leq 171, \quad 2x + 3y \leq 24, \quad x \geq 0, \quad y \geq 0
\]

Would you like more details on how these constraints are derived?

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Here are 5 related questions:
1. What would be the objective function for maximizing profit?
2. How would you graph the feasible region for this problem?
3. Can this problem be solved using linear programming methods?
4. What would be the optimal number of mountain bikes and racing bikes to maximize profit?
5. How do changes in labor hours affect the constraints?

**Tip**: Graphing the constraints helps in visualizing the feasible region for optimization.

A company produces two types of bicycles; mountain bikes and racing bikes. It takes 5 hours of assembly time and 2 hours of mechanical tuning to produce a mountain bike. It takes 8 hours of assembly time and 3 hours of mechanical tuning to produce a racing bike. The company has at most 24 hours of mechanical tuning labor per week and at most 171 hours of assembly labor per week. The company's profit is USD 120 for each mountain bike produced and USD 100 for each racing bike produced. The company wants to make as much money as possible. Let x = the number of mountain bikes they produce, and let y = the number of racing bikes they produce. What are the constraints for this problem?

Explore the linear programming constraints for a company producing mountain and racing bikes. Understand how to apply assembly time and mechanical tuning constraints, and solve using optimization methods to maximize profit. This detailed guide is ideal for students studying operations research or linear programming.

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