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 the number of computers that should be sent from each plant to the retail outlet.
Let's start by defining some variables:
Let x be the number of computers sent from the Rochester plant.
Let y be the number of computers sent from the Queens plant.
To maximize profit, we need to maximize the total profit, which is the sum of the profits from the computers sent from both plants.
The profit from the Rochester plant is calculated as:
Profit_Rochester = x * $40
The profit from the Queens plant is calculated as:
Profit_Queens = y * $30
We have constraints that the total cost of transportation should not exceed $3000 and the total time taken for transportation should not exceed 2100 hours.
The cost of transportation from the Rochester plant is calculated as:
Cost_Rochester = x * $15
The cost of transportation from the Queens plant is calculated as:
Cost_Queens = y * $30
So, the total cost of transportation is:
Total_Cost = Cost_Rochester + Cost_Queens = (x * $15) + (y * $30)
The total time taken for transportation is:
Total_Time = Time_Rochester + Time_Queens = (x * 15 hours) + (y * 20 hours)
Now, we can formulate the profit maximization problem with the given constraints and objective:
Objective function: Maximize Profit = Profit_Rochester + Profit_Queens = (x * $40) + (y * $30)
Subject to the constraints:
1. Total Cost <= $3000: (x * $15) + (y * $30) <= $3000
2. Total Time <= 2100 hours: (x * 15 hours) + (y * 20 hours) <= 2100
3. Non-negativity constraint: x >= 0 and y >= 0
You can solve this problem using techniques such as linear programming or graphical methods to find the optimal values of x and y that maximize the company's profit while satisfying the given constraints.