To minimize the objective function $C = 2x + 5y$ subject to constraints involving "greater than or equal to," we need to evaluate the function $C$ at each of the given vertices and choose the one(s) that yield the smallest value of $C$.

Let's compute $C$ for each of the vertices:

1. Vertex (5, 20): $C = 2(5) + 5(20) = 10 + 100 = 110$

2. Vertex (0, 30): $C = 2(0) + 5(30) = 0 + 150 = 150$

3. Vertex (0, 10): $C = 2(0) + 5(10) = 0 + 50 = 50$

4. Vertex (0, 40): $C = 2(0) + 5(40) = 0 + 200 = 200$

5. Vertex (30, 0): $C = 2(30) + 5(0) = 60 + 0 = 60$

6. Vertex (10, 0): $C = 2(10) + 5(0) = 20 + 0 = 20$

7. Vertex (15, 0): $C = 2(15) + 5(0) = 30 + 0 = 30$

8. Vertex (12, 6): $C = 2(12) + 5(6) = 24 + 30 = 54$

9. Vertex (12, 7): $C = 2(12) + 5(7) = 24 + 35 = 59$

Summary of $C$ values:

• (5, 20): $C = 110$
• (0, 30): $C = 150$
• (0, 10): $C = 50$
• (0, 40): $C = 200$
• (30, 0): $C = 60$
• (10, 0): $C = 20$
• (15, 0): $C = 30$
• (12, 6): $C = 54$
• (12, 7): $C = 59$

Choosing the Vertex with the Minimum $C$:

The vertex that gives the minimum value of $C$ is (10, 0), with $C = 20$.

So, the vertex (10, 0) will be used to minimize the objective function $C = 2x + 5y$ under the given constraints.

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Would you like more details or have any questions about this problem?

Here are some related questions that might help reinforce your understanding:

1. What is the significance of the vertices in linear programming problems?
2. How do you determine which vertex to evaluate in a minimization problem?
3. What are the typical forms of constraints in a linear programming problem?
4. How would the solution change if the objective function were to maximize instead of minimize?
5. What does the slope of the objective function line indicate in a linear programming problem?
6. What is the feasible region, and how is it determined?
7. How does the type of inequality ("greater than or equal to" vs. "less than or equal to") affect the feasible region?
8. Can multiple vertices yield the same minimum value for the objective function? What does this mean for the solution?

Tip: When minimizing or maximizing an objective function in linear programming, always evaluate the function at all vertices of the feasible region to ensure you find the correct optimal solution.