6. A computer company has two manufacturing plants, one in Rochester and one in Queens. Transporting a computer from Rochester to the retail outlet takes 15 hours and costs $15, while transporting a computer from Queens to the retail outlet takes 20 hours and costs $30. The profit on each computer manufactured in Rochester is $40, and the profit on each computer manufactured in Queens is $30. The Rochester plant has 80 computers available, and the Queens plant has 120 computers available. If there are 2100 hours and $3000 allotted for transporting the computers, how many computers should be sent to the retail outlet from each of the two plants to maximize the company’s profit?
To maximize the company's profit, we need to determine how many computers to send from each plant to the retail outlet.
Let's start by setting up some variables:
Let R be the number of computers sent from Rochester.
Let Q be the number of computers sent from Queens.
To transport R computers from Rochester to the retail outlet, it will take 15R hours and cost 15R dollars.
To transport Q computers from Queens to the retail outlet, it will take 20Q hours and cost 30Q dollars.
Now, let's consider the constraints:
1. The total transportation time cannot exceed 2100 hours: 15R + 20Q ≤ 2100.
2. The total transportation cost cannot exceed $3000: 15R + 30Q ≤ 3000.
3. The number of computers sent from Rochester cannot exceed 80: R ≤ 80.
4. The number of computers sent from Queens cannot exceed 120: Q ≤ 120.
The objective is to maximize the profit:
Profit = 40R + 30Q.
Now, we can solve this linear programming problem using the graphical method or linear programming software.
Alternatively, we can use the concept of the corner points of the feasible region to find the optimal solution.
Step 1: Find the corner points of the feasible region.
To find the corner points, we can solve the system of inequalities:
15R + 20Q = 2100 (transportation time constraint)
15R + 30Q = 3000 (transportation cost constraint)
R ≤ 80 (Rochester constraint)
Q ≤ 120 (Queens constraint)
The corner points of the feasible region are:
(80, 0), (0, 100), (60, 60), (80, 45), (80, 0).
Step 2: Calculate the profit at each corner point.
Profit (80, 0) = 40(80) + 30(0) = $3200
Profit (0, 100) = 40(0) + 30(100) = $3000
Profit (60, 60) = 40(60) + 30(60) = $4200
Profit (80, 45) = 40(80) + 30(45) = $3900
Profit (80, 0) = $3200
Step 3: Find the maximum profit.
The maximum profit is $4200 when 60 computers are sent from Rochester and 60 computers are sent from Queens.