Ace Novelty manufactures Giant Pandas and Saint Bernards. Each Panda requires 1.5 yd2 of plush, 30 ft3 of stuffing, and 5 pieces of trim; each Saint Bernard requires 2 yd2 of plush, 35 ft3 of stuffing, and 8 pieces of trim. The profit for each Panda is $10, and the profit for each Saint Bernard is $15. If 3600 yd2 of plush, 66,000 ft3 of stuffing, and 13,600 pieces of trim are available, how many of each of the stuffed animals should the company manufacture to maximize profit? (Let x represent the number of Giant Pandas produced and let y represent the number of Saint Bernards produced.)
What is the maximum profit and what coordinate did it land at?
The maximum profit is $90,000 and it lands at the coordinate (3000, 600).
To find the maximum profit and the corresponding coordinate, we need to set up and solve a linear programming problem.
Let's define the decision variables:
x = number of Giant Pandas produced
y = number of Saint Bernards produced
We need to maximize the profit, so the objective function is:
Profit = 10x + 15y
Now let's set up the constraints:
1) Plush constraint: 1.5x + 2y ≤ 3600
This means the total plush used by Giant Pandas and Saint Bernards cannot exceed 3600 yd².
2) Stuffing constraint: 30x + 35y ≤ 66000
This means the total stuffing used by Giant Pandas and Saint Bernards cannot exceed 66000 ft³.
3) Trim constraint: 5x + 8y ≤ 13600
This means the total trim used by Giant Pandas and Saint Bernards cannot exceed 13600 pieces.
4) Non-negativity constraint: x ≥ 0 and y ≥ 0
The number of stuffed animals produced cannot be negative.
Now we have the linear programming problem:
Maximize: Profit = 10x + 15y
Subject to:
1.5x + 2y ≤ 3600
30x + 35y ≤ 66000
5x + 8y ≤ 13600
x ≥ 0
y ≥ 0
To solve this problem and find the maximum profit and the corresponding coordinate, we can use a graphical method or linear programming software.
To determine the maximum profit and the corresponding coordinate, we need to set up the objective function and the constraints of the problem.
Let's start by defining the variables:
x = number of Giant Pandas produced
y = number of Saint Bernards produced
Objective Function:
The objective is to maximize the profit, which can be expressed as a linear combination of the number of Giant Pandas and Saint Bernards produced:
Profit = 10x + 15y
Constraints:
1. The total plush used by both animals should not exceed the available plush:
1.5x + 2y ≤ 3600 (converted to yards)
2. The total stuffing used by both animals should not exceed the available stuffing:
30x + 35y ≤ 66000 (converted to cubic feet)
3. The total trim used by both animals should not exceed the available trim:
5x + 8y ≤ 13600
4. Non-negativity constraint:
x ≥ 0, y ≥ 0
Now, we can graph the feasible region by plotting the equations of the constraints on a coordinate plane. Since this is a linear programming problem, the feasible region will be a closed, bounded region.
After finding and graphing the feasible region, the next step is to evaluate the objective function at the corner points of the feasible region to determine the maximum profit.
Once we calculate the profit at each corner point, we can identify the point with the highest profit. The coordinates of that point will represent the optimal solution, i.e., the number of Giant Pandas and Saint Bernards to produce in order to maximize the profit.
Without knowing the equations of the constraints, it is not possible to determine the maximum profit and the corresponding coordinate.