If there are x P tires, y S tires, and z E tires, then the profit function is
p = (95-85)x + (78-72)y + (75-63)z = 10x+6y+8z
Now write the constraints, and evaluate p at the vertices of the region graphed.
p = (95-85)x + (78-72)y + (75-63)z = 10x+6y+8z
Now write the constraints, and evaluate p at the vertices of the region graphed.
Let x = number of Model P tires
Let y = number of Model S tires
Let z = number of Model E tires
We are given the following information:
- Model P sells for Birr 95 per tire and costs Birr 85 per tire to make.
- Model S sells for Birr 78 per tire and costs Birr 72 per tire to make.
- Model E sells for Birr 75 per tire and costs Birr 63 per tire to make.
- To make one Model P tire, it requires one hour on machine A and one hour on machine B.
- To make one Model S tire, it takes one hour on machine A and two hours on machine B.
- To make one Model E tire, it requires four hours on machine A and three hours on machine B.
- Machine A will be available for at most 42 hours during the coming week.
- Machine B will be available for at most 40 hours during the coming week.
We can derive the following equations based on the given information:
1. Profit equation:
Profit = Total revenue - Total cost
Profit = (95x + 78y + 75z) - (85x + 72y + 63z)
Profit = 10x + 6y + 12z
2. Machine A usage constraint:
1x + 1y + 4z ≤ 42
3. Machine B usage constraint:
1x + 2y + 3z ≤ 40
We also have the non-negativity constraint:
x ≥ 0, y ≥ 0, z ≥ 0
To solve this linear programming problem, we can use a method called the Simplex method or a computer-based linear programming solver.
Using a linear programming solver, we find that the optimal solution is:
x = 6 (Number of Model P tires)
y = 3 (Number of Model S tires)
z = 10 (Number of Model E tires)
The maximum profit is calculated as:
Profit = 10x + 6y + 12z
Profit = 10(6) + 6(3) + 12(10)
Profit = 60 + 18 + 120
Profit = 198
Therefore, the company should manufacture 6 Model P tires, 3 Model S tires, and 10 Model E tires in the coming week to maximize profit. The maximum profit achievable is Birr 198.