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. 

The problem is about maximizing the profit of Bata Aerobics Company based on two models of steppers: luxury and standard. Let's define the variables:

- $ x $: number of luxury steppers
- $ y $: number of standard steppers

### Given Data:
- **Luxury Stepper**:
  - Requires 20 lbs. of plastic and 9 minutes of labor.
  - Profit per unit: $40.

- **Standard Stepper**:
  - Requires 30 lbs. of plastic and 6 minutes of labor.
  - Profit per unit: $15.

- **Constraints**:
  - Total plastic available per day: 12,600 lbs.
  - Total labor available per day: 60 hours = 3,600 minutes (since $60 \times 60 = 3,600$).

### Step 1: Objective Function

The objective is to **maximize the total profit**.

The profit is calculated by adding the profit from luxury and standard steppers:
\[
\text{Profit} = 40x + 15y
\]

Thus, the **objective function** to maximize is:
\[
P = 40x + 15y
\]
where:
- $ x $ is the number of luxury steppers,
- $ y $ is the number of standard steppers.

### Step 2: Constraints
Now, let's set up the constraints based on the resources:

1. **Plastic Constraint**:
   The total amount of plastic used for both models cannot exceed 12,600 lbs:
   \[
   20x + 30y \leq 12,600
   \]

2. **Labor Constraint**:
   The total labor time for both models cannot exceed 3,600 minutes:
   \[
   9x + 6y \leq 3,600
   \]

3. **Non-Negativity**:
   The number of steppers cannot be negative:
   \[
   x \geq 0, \quad y \geq 0
   \]

---

So, the problem is to **maximize** the objective function:
\[
P = 40x + 15y
\]
subject to the constraints:
\[
20x + 30y \leq 12,600
\]
\[
9x + 6y \leq 3,600
\]
\[
x \geq 0, \quad y \geq 0
\]

Would you like to proceed with solving this linear programming problem or need any further clarification?

---

Here are 5 related questions for further exploration:

1. How do we graphically solve this linear programming problem?
2. What are the corner points of the feasible region?
3. How do we determine the maximum profit using the objective function?
4. Can sensitivity analysis be performed on the constraints?
5. How would the solution change if the available plastic increased?

**Tip**: Always ensure to clearly define your variables and constraints when setting up linear programming problems.

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.

Explore how to maximize profits for Bata Aerobics Company by solving a linear programming problem. Learn how to set up the objective function and constraints, and understand key mathematical concepts such as linear programming and optimization. Suitable for students in Grades 10-12.

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