LINEAR PROGRAMMING

Mrs. Bear has a home business where she makes and sells stuffed animals, primarily dolphins and manatees.

Each dolphin requires:
- 2 square feet of material
- 4 ounces of stuffing
- 3 hours of labor

Each manatee requires:
- 4 square feet of material
- 8 ounces of stuffing
- 1 hour of labor

Resources available each week:
- 80 square feet of material
- 120 ounces of stuffing
- 60 hours of labor

Profits:
- $5 per dolphin
- $7 per manatee

Objective:
Determine how many dolphins and manatees Mrs. Bear should make to maximize her profit, and calculate the maximum profit.

Questions:
a. What is the value of an additional hour of labor to the total profit?
b. How much would the profit increase with an additional ounce of stuffing?
c. How much of the 80 square feet of material are actually used in the optimal solution?
d. What is the lowest profit for dolphins that would result in the same solution?
e. What is the highest profit for manatees that would result in the same solution?

To solve this linear programming problem, we can use the simplex method to determine the optimal solution and maximum profit for Mrs. Bear's stuffed animal business.

Explanation:

To solve this linear programming problem, we can use the method of linear programming called the simplex method. Let's define two variables, x and y, to represent the number of dolphins and manatees Mrs. Bear should make, respectively.

The objective function is to maximize the profit, which can be expressed as 5x + 7y. The constraints are:

2x + 4y ≤ 80 (material constraint)
4x + 8y ≤ 120 (stuffing constraint)
3x + y ≤ 60 (labor constraint)
x ≥ 0, y ≥ 0 (non-negativity constraint)

By solving this linear programming problem using the simplex method, we can determine the optimal solution, which represents the number of dolphins and manatees Mrs. Bear should make to maximize her profit, as well as the maximum profit.

Learn more about Linear Programming here:

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