An electronics company manufactures personal tape players on which it makes a profit of $28 each and personal CD players where the profit is $33 each. The company intends to produce at least 60 tape players and 100 CD players per day but its factory is not equipped to assemble more than a total of 200 of these items a day. What should the company’s daily production of these items be if profits are to be as great as possible?
Linear Programming??????
Yes, this is a linear programming problem. Linear programming is a mathematical optimization technique used to find the maximum or minimum value of a linear objective function, subject to a set of linear constraints.
In this case, we can formulate the problem as follows:
Let x be the number of tape players produced per day.
Let y be the number of CD players produced per day.
Objective function: Maximize Profit = 28x + 33y
Constraints:
- x ≥ 60 (Minimum production of tape players)
- y ≥ 100 (Minimum production of CD players)
- x + y ≤ 200 (Total production limit)
To find the maximum profit, we need to solve this linear programming problem by graphing or using a linear programming solver software.
Yes, this problem can be solved using linear programming, which is a mathematical technique used to determine the optimal solution for a set of constraints and an objective function. In this case, the objective is to maximize the profit, while the constraints are the minimum production requirements and the maximum production capacity.
To solve this problem using linear programming, we can define two variables:
Let x be the number of tape players produced per day.
Let y be the number of CD players produced per day.
The objective function can be defined as the total profit:
P = 28x + 33y
Now, let's consider the constraints:
Constraint 1: The company intends to produce at least 60 tape players per day:
x ≥ 60
Constraint 2: The company intends to produce at least 100 CD players per day:
y ≥ 100
Constraint 3: The total production capacity of the factory is limited to 200 items per day:
x + y ≤ 200
Since we are looking to maximize the profit, this problem can be solved using a method called the graphical method. We can graph the constraints and find the feasible region (the region in which all constraints are satisfied). The point within this region that provides the maximum profit will be the optimal solution.
By graphing the constraints, we find that the feasible region is a triangular region bounded by the lines x = 60, y = 100, and x + y = 200.
To find the optimal solution, we need to evaluate the objective function at the corner points of the feasible region. These corner points are (60, 100), (60, 140), and (140, 60).
By substituting these values into the objective function, we find the profit at each corner point:
P1 = 28(60) + 33(100) = $6180
P2 = 28(60) + 33(140) = $8460
P3 = 28(140) + 33(60) = $8540
Therefore, the maximum profit is $8540, which occurs when the company produces 140 tape players and 60 CD players per day.