To solve the problem, we will evaluate the objective function $z = 5x + 3y$ at each vertex of the feasible region, which are points $A(2,10)$, $B(7,5)$, $C(8,3)$, and $D(1,2)$, and then find the maximum and minimum values.

Step-by-Step Solution

1. Calculate $z$ at each point:

   • Point A (2,10):
     $z = 5(2) + 3(10) = 10 + 30 = 40$

   • Point B (7,5):
     $z = 5(7) + 3(5) = 35 + 15 = 50$

   • Point C (8,3):
     $z = 5(8) + 3(3) = 40 + 9 = 49$

   • Point D (1,2):
     $z = 5(1) + 3(2) = 5 + 6 = 11$

2. Answering Each Part:

   • Part A: The values of the objective function $z$ at each corner:

     • At $A(2,10)$: $z = 40$
     • At $B(7,5)$: $z = 50$
     • At $C(8,3)$: $z = 49$
     • At $D(1,2)$: $z = 11$

   • Part B: The maximum value of $z$ is 50 at point $B(7,5)$.

   • Part C: The minimum value of $z$ is 11 at point $D(1,2)$.

Would you like further details or explanations on any step?

Related Questions

1. How do you determine the feasible region in a linear programming problem?
2. What is the significance of the vertices of the feasible region in optimization?
3. How does changing the objective function coefficients impact the maximum and minimum values?
4. Can there be multiple maximum or minimum points in a linear programming problem?
5. What are some applications of linear programming in real-life scenarios?

Tip

When optimizing a linear function subject to constraints, always evaluate the objective function at the vertices of the feasible region.