Question

A toy manufacturer makes lightweight balls for indoor play. The large basketball uses 4 ounces of foam and 20 minutes of labor and brings a profit of ​$2.50. The football uses 3 ounces of foam and 30 minutes of labor and brings a profit of ​$2. The manufacturer has available 43.5 pounds of foam and 110 ​labor-hours a week. Use the simplex method to determine the optimal production schedule so as to maximize profits. Show that the geometric method gives the same solution.

Answer 1

To determine the optimal production schedule to maximize profits, we need to create a linear programming model using the simplex method.

Step-by-step explanation:

To determine the optimal production schedule to maximize profits, we need to create a linear programming model using the simplex method. Let's define our decision variables as follows:

• x = number of large basketballs to produce
• y = number of footballs to produce

Our objective is to maximize the profit, so our objective function can be written as:

Z = 2.5x + 2y

We have the following constraints:

• 4x + 3y ≤ 43.5 (foam constraint)
• 20x + 30y ≤ 110 (labor constraint)
• x, y ≥ 0 (non-negativity constraint)

To solve this linear programming problem with the simplex method, we will use a software or calculator that supports linear programming.