The problem you're working on deals with linear programming, specifically finding the maximum and minimum values of the given objective functions on the region of feasible solutions shown on the graph.

Step-by-Step Approach:

1. Objective Function:

   • You're asked to evaluate the objective function $z = 1.75x + 0.75y$.

2. Feasible Region:

   • The graph shows a region where the solution is feasible, marked by points: $(0, 0)$, $(8, 3)$, $(5, 7)$, and $(0, 10)$. These are the vertices of the polygonal region, and you must evaluate the objective function at each of these vertices to find the maximum value.

3. Evaluation of the Objective Function: Let's evaluate $z = 1.75x + 0.75y$ at each of the vertices:

   • At $(0, 0)$:
     $z = 1.75(0) + 0.75(0) = 0$
   • At $(8, 3)$:
     $z = 1.75(8) + 0.75(3) = 14 + 2.25 = 16.25$
   • At $(5, 7)$:
     $z = 1.75(5) + 0.75(7) = 8.75 + 5.25 = 14$
   • At $(0, 10)$:
     $z = 1.75(0) + 0.75(10) = 7.5$

4. Maximum Value:

   • The maximum value of $z$ occurs at $(8, 3)$ and is $16.25$.

Thus, the maximum value of the objective function $z = 1.75x + 0.75y$ is 16.25 at the point $(8, 3)$.

Answer:

You should select option A:

• The maximum value of the objective function is $16.25$ and is located only at the point $(8, 3)$.

Would you like more details on linear programming or have further questions?

Additional Questions:

1. How do you calculate the minimum value of the objective function?
2. Can the maximum value occur along a line segment in some cases?
3. What are the conditions for a point to be a feasible solution in linear programming?
4. What is the significance of the vertices in finding the optimal solution in linear programming?
5. How would the problem change if the objective function were different?

Tip:

In linear programming, the maximum and minimum values of the objective function always occur at one of the vertices of the feasible region.