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To solve part (a) and find the maximum value of \( z = 0.40x + 0.25y \) based on the given graph, we can evaluate the objective function at the corner points of the feasible region, which are:

1. \( (0,0) \)
2. \( (7,0) \)
3. \( (6,4) \)
4. \( (3,8) \)
5. \( (0,11) \)

We will compute the value of \( z \) for each of these points:

1. At \( (0,0) \):
   \[
   z = 0.40(0) + 0.25(0) = 0
   \]
2. At \( (7,0) \):
   \[
   z = 0.40(7) + 0.25(0) = 2.8
   \]
3. At \( (6,4) \):
   \[
   z = 0.40(6) + 0.25(4) = 2.4 + 1 = 3.4
   \]
4. At \( (3,8) \):
   \[
   z = 0.40(3) + 0.25(8) = 1.2 + 2 = 3.2
   \]
5. At \( (0,11) \):
   \[
   z = 0.40(0) + 0.25(11) = 0 + 2.75 = 2.75
   \]

The maximum value of \( z \) occurs at the point \( (6,4) \), where \( z = 3.4 \).

### Answer:
The maximum value of the objective function \( z = 0.40x + 0.25y \) is **3.4** and it is located at the point **(6,4)**.

Would you like more details on any step or further clarification?

Here are 5 related questions to expand on this topic:
1. What is the minimum value of \( z = 0.40x + 0.25y \) within the feasible region?
2. How do you find the feasible region for linear programming problems?
3. What happens if the objective function has no maximum or minimum values?
4. Can we use another method, like the Simplex method, to solve this problem?
5. How would the answer change if the objective function coefficients were different?

**Tip:** Always check the corner points of the feasible region when solving linear programming problems, as the maximum and minimum values always occur at one of these points.

What is the maximum of z = 0.40x + 0.25y?

Find the maximum value of the objective function z = 0.40x + 0.25y using the vertices of the feasible region. This problem covers linear programming concepts and explores evaluating objective functions at corner points. Suitable for high school students in Grades 10-12.

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