Argus company makes three products: A, B, and C. Each unit of A costs $4, each unit of B costs $2, and each unit of C costs $1 to produce. Argus must produce at least 20 As, 30 Bs, and 40 Cs combined. Minimize Argus's costs
To minimize Argus's costs, we need to determine how many units of A, B, and C should be produced to meet the required quantities while minimizing the total cost.
Let's assume Argus produces x units of A, y units of B, and z units of C.
Based on the given information, we have the following cost equations:
Cost of A = 4x
Cost of B = 2y
Cost of C = 1z
Now, we need to consider the production constraints. Argus must produce at least 20 units of A, 30 units of B, and 40 units of C combined. Therefore, we have the following constraints:
x ≥ 20
y ≥ 30
z ≥ 40
To minimize the total cost, we need to minimize the expression:
Total Cost = Cost of A + Cost of B + Cost of C
= 4x + 2y + z
Now, we can solve this problem using linear programming techniques. One way to solve it is through graphical method. Here's how you can do it:
1. Draw the graph for each constraint: x ≥ 20, y ≥ 30, and z ≥ 40.
- Draw the line x = 20 (vertical line passing through x = 20)
- Draw the line y = 30 (horizontal line passing through y = 30)
- Draw the line z = 40 (horizontal line passing through z = 40)
- Shade the feasible region (the area where all the constraints are satisfied).
2. Calculate the vertices (corners) of the feasible region by finding the intersection points of the lines representing the constraints.
3. Substitute the coordinates of each vertex into the Total Cost expression (4x + 2y + z) to determine the cost at each vertex.
4. Identify the vertex with the minimum total cost. This vertex represents the optimal solution, where Argus can minimize their costs.
By following these steps, you can find the optimal solution and determine the specific quantities of A, B, and C that minimize Argus's costs.