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The objective function in linear programming is like a formula that tells us what we want to achieve. In the case of Bata Aerobics, the aim is to maximize profit. Hence, the objective function represents how much money we can make based on the number of luxury and standard steppers produced.

For this problem, each luxury stepper sold brings a profit of \(40, and each standard stepper yields \)30. Thus, the objective function, denoted as \( P \) (standing for profit), is expressed as \( P = 40x + 30y \), where \( x \) is the number of luxury steppers and \( y \) is the number of standard steppers.

By using this objective function, we can calculate the total profit for any combination of \( x \) and \( y \). The goal is to find values of \( x \) and \( y \) that maximize the profit, subject to given constraints.

Constraints are the limitations or restrictions that impact decision-making in linear programming. They define the feasible options available. In this context, Bata Aerobics faces constraints regarding the amount of resources available: plastic and labor.

There are two main constraints to consider:

• Plastic Constraint: Each luxury model requires 10 lbs, and each standard model needs 16 lbs of plastic. With only 6000 lbs available, the constraint can be written as \( 10x + 16y \leq 6000 \).
• Labor Constraint: Each luxury model demands 10 minutes, and each standard model takes 8 minutes of labor. Assuming 60 labor hours (or 3600 minutes) are available each day, the constraint can be expressed as \( 10x + 8y \leq 3600 \).

Furthermore, there are non-negativity constraints which state that the number of steppers produced cannot be negative, hence \( x \geq 0 \) and \( y \geq 0 \). Collectively, these constraints form the boundaries within which production must occur.

The feasible region represents all possible combinations of luxury and standard steppers that can be produced, considering the constraints. It's like the "playground" of options we have.

To identify the feasible region, you plot each constraint on a graph. For example, by plotting the lines \( 10x + 16y = 6000 \) and \( 10x + 8y = 3600 \), you find their intersecting points and shade the region where all constraints overlap.

The non-negativity constraints ensure this region is bounded in the first quadrant, as both \( x \) and \( y \) must be non-negative. Within this feasible region lie potential solutions to the problem, and the optimal solution will always sit at one of the vertices (corners) of this area.

The optimal solution is the best choice among the feasible options, where the objective function reaches its highest value. In our exercise with Bata Aerobics, this means finding the combination of steppers that results in the maximum profit.

After graphing the feasible region, we pinpoint the vertices, which are calculated points where constraints intersect or where constraints meet the axes. For this scenario, the vertices were at: \((120, 300)\), \((600, 0)\), and \((0, 375)\). By substituting these points into the objective function \( P = 40x + 30y \), we assess the profit potential at each.

The calculations showed that the maximum profit of $24,000 was achieved at the vertex \((600, 0)\), meaning the best strategy for Bata would be to produce 600 luxury steppers and no standard steppers daily. Thus, the optimal solution directs decision-making by confirming which production strategy brings the greatest return.