The corner point of the feasible solutions are(0,0) (3,0)(2,1)(0,7/3) the maximum value of Z= 4x+5y is

Options

A. 12
B. 13
C. 35/3
D. 0

Answer

To find the maximum value of \( Z = 4x + 5y \) at the corner points of the feasible solutions, we will evaluate the function at each of the given corner points: \( (0,0) \), \( (3,0) \), \( (2,1) \), and \( (0,\frac{7}{3}) \).

### Step-by-Step Solution:

1. **Evaluate \( Z \) at \( (0,0) \)**:
\[ Z = 4(0) + 5(0) = 0 \]

2. **Evaluate \( Z \) at \( (3,0) \)**:
\[ Z = 4(3) + 5(0) = 12 \]

3. **Evaluate \( Z \) at \( (2,1) \)**:
\[ Z = 4(2) + 5(1) = 8 + 5 = 13 \]

4. **Evaluate \( Z \) at \( (0,\frac{7}{3}) \)**:
\[ Z = 4(0) + 5\left(\frac{7}{3}\right) = \frac{35}{3} \approx 11.67 \]

5. **Compare the values of \( Z \)**:
- At \( (0,0) \), \( Z = 0 \)
- At \( (3,0) \), \( Z = 12 \)
- At \( (2,1) \), \( Z = 13 \)
- At \( (0,\frac{7}{3}) \), \( Z \approx 11.67 \)

6. **Determine the maximum value**:
The maximum value of \( Z \) occurs at the point \( (2,1) \) where \( Z = 13 \).

### Final Answer:
The maximum value of \( Z = 4x + 5y \) is **13**.

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