The graph to the right shows a region of feasible solutions. Use this region to find maximum and minimum values of the given objective​ functions, and the locations of these values on the graph.
​(a) zequals0.40xplus0.25y
​(b) zequals1.50xplus0.50y                                                                                                                       
     (b) What is the maximum of zequals1.50xplus0.50​y? Select the correct answer below​ and, if​ necessary, fill in the answer boxes to complete your choice.
A.
The maximum value of the objective function zequals1.50xplus0.50y is 
  
enter your response here and is located at 
  
enter your response here.
​(Type an exact​ answer.)
B.
The maximum does not exist.

To solve this problem, we need to maximize the given objective function \( z = 1.50x + 0.50y \) within the region of feasible solutions determined by the graph, typically defined by a set of constraints.

### Steps to Maximize \( z = 1.50x + 0.50y \):
1. **Identify the Feasible Region**: The region of feasible solutions is where all constraints (usually inequalities) are satisfied. These constraints are typically defined by lines or boundaries on the graph. The feasible region will be a polygon or possibly unbounded, depending on the graph.
   
2. **Find Corner Points**: The maximum and minimum values of a linear objective function over a feasible region occur at the vertices (corner points) of the polygon. So, we need to determine the coordinates of these points by analyzing the intersections of the constraint lines.

3. **Evaluate the Objective Function**: Once the corner points (vertices) are identified, substitute the coordinates of each vertex into the objective function \( z = 1.50x + 0.50y \).

4. **Compare Values**: The largest value of \( z \) at these corner points will be the maximum, and the smallest will be the minimum.

### Answer Choices:
- **A**: The maximum value exists, and it is located at specific coordinates.
- **B**: The maximum does not exist if the feasible region is unbounded in the direction that maximizes \( z \).

To move forward, I need either:
- The coordinates of the vertices from the graph or
- A description of the feasible region, including any constraints or intersection points.

Once I have that information, I can help find the maximum value of \( z = 1.50x + 0.50y \) and its location.

Would you like to provide the graph or the vertices of the feasible region?

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Here are 5 follow-up questions you might find useful:
1. What are the constraints that define the feasible region?
2. Can you provide the coordinates of the vertices in the feasible region?
3. Is the region bounded or unbounded?
4. What method are you using to find the vertices, if any?
5. Do you need assistance calculating the values of \( z \) at each vertex?

**Tip:** The maximum or minimum of a linear objective function always occurs at one of the vertices of the feasible region.

The graph to the right shows a region of feasible solutions. Use this region to find maximum and minimum values of the given objective​ functions, and the locations of these values on the graph. (a) z = 0.40x + 0.25y (b) z = 1.50x + 0.50y. What is the maximum of z = 1.50x + 0.50​y? Select the correct answer below​ and, if​ necessary, fill in the answer boxes to complete your choice. A. The maximum value of the objective function z = 1.50x + 0.50y is enter your response here and is located at enter your response here. ​(Type an exact​ answer.) B. The maximum does not exist.

Solve a linear programming problem by maximizing the objective function z = 1.50x + 0.50y within a feasible region. This tutorial explains key concepts such as identifying corner points, evaluating the objective function, and finding maximum and minimum values. Suitable for Grades 10-12.

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