Let's go through the problem step by step.

Problem Breakdown:

• Variables:

  • $x$: Number of Luxury Steppers.
  • $y$: Number of Standard Steppers.

• Given Resources:

  • Plastic available: 12,600 lbs.
  • Labor available: 60 hours (which is $60 \times 60 = 3,600$ minutes).

• Resource Usage:

  • Each Luxury Stepper requires 20 lbs of plastic and 9 minutes of labor.
  • Each Standard Stepper requires 30 lbs of plastic and 6 minutes of labor.

• Profit:

  • Profit from each Luxury Stepper: $40.
  • Profit from each Standard Stepper: $15.

Step 1: Write the Objective Function

The goal is to maximize the profit $P$, which is given by: $P = 40x + 15y$

So, the correct option for Question 9 is:

• Answer: (e) Max P = 40x + 15y

Step 2: Write the Constraints

• Plastic Constraint: $20x + 30y \leq 12,600$

• Labor Constraint: $9x + 6y \leq 3,600$

• Non-negativity Constraints: $x \geq 0, \quad y \geq 0$

Step 3: Solve the Linear Programming Problem

To solve this linear programming problem, we would typically use the graphical method or simplex method. Here, I'll outline the key points and the results.

1. Convert labor constraint to standard form: $3x + 2y \leq 1,200$

2. Solve the system of inequalities by evaluating at the corners of the feasible region.

Feasible Points:

• $x = 0, y = 420$: $P = 40(0) + 15(420) = 6,300$
• $x = 400, y = 0$: $P = 40(400) + 15(0) = 16,000$
• $x = 216, y = 276$: $P = 40(216) + 15(276) = 14,500$

Step 4: Find the Maximum Profit

The optimal solution is to produce 400 Luxury Steppers and 0 Standard Steppers, resulting in a maximum profit of $16,000.

• Answer to Question 10: (b) $16,000

• Answer to Question 11: (d) 400 Luxury and 0 Standard Steppers

Summary of Answers:

• Question 9: (e) Max P = 40x + 15y
• Question 10: (b) $16,000
• Question 11: (d) 400 Luxury and 0 Standard Steppers

Would you like further details or explanations on any of these steps?

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Related Questions:

1. What is the graphical method for solving linear programming problems?
2. How do you convert labor constraints from hours to minutes in linear programming problems?
3. What is the significance of non-negativity constraints in linear programming?
4. How can the Simplex Method be applied to this problem?
5. What are the differences between feasible and optimal solutions in linear programming?

Tip: When dealing with resource constraints in linear programming, always ensure the units match across all constraints for accurate calculations.