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.