Use the following problem to answer questions 9 – 11.
The Bata Aerobics Company manufactures two models of steppers used for aerobics exercises. To
manufacture each luxury model requires 20 lbs. of plastic and 9 min of labor. To manufacture each
standard model requires 30 lbs. of plastic and 6 min of labor. The profit for each luxury model is $40,
and the profit for each standard model is $15. If 12,600 lbs. of plastic and 60 hrs. of labor are available
for the production of the steppers per day, how many steppers of each model should Bata produce in
order to maximize its profits? Let x = number of Luxury Steppers and y = number of Standard Steppers.
9. Give the objective function.
a. Min P = 20x + 9y
b. Min P = 20x + 30y
c. Max P = 9x + 6y
d. Min P = 30x + 6y
e. Max P = 40x + 15y
10. Find the optimal value of the objective function.
a. $14,500
b. $16,000
c. $11,283
d. $12,780
e. $19,800
11. How many of each stepper model should they produce?
a. 0 Luxury and 600 Standard Steppers
b. 0 Luxury and 420 Standard Steppers
c. 216 Luxury and 276 Standard Steppers
d. 400 Luxury and 0 Standard Steppers
e. 630 Luxury and 0 Standard Steppers 

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?

---

**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.

Use the following problem to answer questions 9 – 11. The Bata Aerobics Company manufactures two models of steppers used for aerobics exercises. To manufacture each luxury model requires 20 lbs. of plastic and 9 min of labor. To manufacture each standard model requires 30 lbs. of plastic and 6 min of labor. The profit for each luxury model is $40, and the profit for each standard model is $15. If 12,600 lbs. of plastic and 60 hrs. of labor are available for the production of the steppers per day, how many steppers of each model should Bata produce in order to maximize its profits?

Solve the Bata Aerobics Company's stepper manufacturing problem using linear programming. Learn how to maximize profits by optimizing the production of luxury and standard steppers with plastic and labor constraints. Includes key formulas and concepts related to optimization, inequalities, and feasible regions. Suitable for Grades 10-12.

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