To find the minimum value of C = 10x + 3y subject to the given constraints, we need to analyze the constraints and find the feasible region.
The constraints are:
1. -x + y ≥ 20
2. 2x + y ≥ 24
3. 2x + y ≤ 13
4. y ≥ 20
To graphically represent these constraints, we can plot the lines and shade the region that satisfies all the constraints.
Let's start by graphing the lines -x + y = 20 and 2x + y = 24:
For -x + y = 20, when x = 0, y = 20, and when y = 0, x = -20. So, we have two points: (0, 20) and (-20, 0). Drawing a line through these points will represent the constraint.
For 2x + y = 24, when x = 0, y = 24, and when y = 0, x = 12. So, we have two points: (0, 24) and (12, 0). Drawing a line through these points will represent the constraint.
Next, let's shade the region that satisfies all the constraints:
- The region above the line -x + y = 20 (including the line itself).
- The region above the line 2x + y = 24 (including the line itself).
- The region below the line 2x + y = 13 (including the line itself).
- The region above the line y = 20 (including the line itself).
Now, to find the minimum value of C = 10x + 3y within this feasible region, we need to evaluate C at each vertex of the shaded region and find the minimum value.
Since I can't see the graph or the exact vertices, I can't provide the exact minimum value of C. But you can use this method to find it by evaluating C at each vertex.