https://file.aiquickdraw.com/mathbot/file/1730080930357adj03whj.jpg

Let's verify the feasible region based on the constraints provided in the problem:

### Problem and Constraints:
1. **Objective function**: \( z = 4.5x + 4y \)
2. **Constraints**:
   - \( x \geq 0 \)
   - \( y \geq 0 \)
   - \( 4x + y \geq 50 \)
   - \( 2x + y \geq 40 \)

### Analyzing the Constraints:

Let's rewrite and plot each inequality constraint:

1. **\( x \geq 0 \)** and **\( y \geq 0 \)**:
   - These constraints indicate that the feasible region will lie in the first quadrant.

2. **\( 4x + y \geq 50 \)**:
   - Rewrite as \( y \geq -4x + 50 \).
   - This line crosses the y-axis at \( (0, 50) \) and the x-axis at \( (12.5, 0) \).

3. **\( 2x + y \geq 40 \)**:
   - Rewrite as \( y \geq -2x + 40 \).
   - This line crosses the y-axis at \( (0, 40) \) and the x-axis at \( (20, 0) \).

### Observing the Graph:

In the image, it appears that the feasible region (bounded by the lines and within the first quadrant) has been shaded. This region should meet all the constraints above, primarily falling above the lines \( y = -4x + 50 \) and \( y = -2x + 40 \), and within the first quadrant. 

### Conclusion:
The graph in the image looks correct as it seems to satisfy all the constraints. The shaded region meets the non-negativity constraints and lies above the lines formed by the inequalities \( 4x + y \geq 50 \) and \( 2x + y \geq 40 \). This appears to be the correct feasible region.

Would you like to confirm any specific points or need further details on verifying constraints?

---

Here are five additional questions you may find useful:
1. How do we identify corner points for optimizing \( z \) in this feasible region?
2. What method would you use to maximize or minimize the objective function \( z \)?
3. How does changing the coefficients in the objective function affect the optimal solution?
4. What would happen to the feasible region if one of the inequalities were changed?
5. How do you know if a feasible region is unbounded?

**Tip:** Always ensure each inequality constraint is transformed correctly for graphing by isolating \( y \) or \( x \) when necessary.

Verify the feasible region for a linear programming problem with the objective function z = 4.5x + 4y and constraints x ≥ 0, y ≥ 0, 4x + y ≥ 50, and 2x + y ≥ 40. Explore how to graph inequalities and identify the feasible region in the first quadrant, satisfying all constraints.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation