If the objective function for an L.P.P. is `Z=3x+4y` and the corner points for unbounded feasible region are (9, 0), (4, 3), (2, 5), and (0, 8), then the maximum value of Z occurs at

Options

A. (0, 8)
B. (2, 5)
C. (4, 3)
D. (9, 0)

Answer

To solve the given Linear Programming Problem (LPP) and find the maximum value of the objective function \( Z = 3x + 4y \) at the specified corner points, we will follow these steps:

### Step 1: Identify the Corner Points
The corner points given are:
1. \( (9, 0) \)
2. \( (4, 3) \)
3. \( (2, 5) \)
4. \( (0, 8) \)

### Step 2: Calculate the Value of Z at Each Corner Point
We will substitute each corner point into the objective function \( Z = 3x + 4y \).

1. **For the corner point \( (9, 0) \)**:
\[ Z = 3(9) + 4(0) = 27 + 0 = 27 \]

2. **For the corner point \( (4, 3) \)**:
\[ Z = 3(4) + 4(3) = 12 + 12 = 24 \]

3. **For the corner point \( (2, 5) \)**:
\[ Z = 3(2) + 4(5) = 6 + 20 = 26 \]

4. **For the corner point \( (0, 8) \)**:
\[ Z = 3(0) + 4(8) = 0 + 32 = 32 \]

### Step 3: Compare the Values of Z
Now we will compare the values of \( Z \) calculated at each corner point:
- At \( (9, 0) \), \( Z = 27 \)
- At \( (4, 3) \), \( Z = 24 \)
- At \( (2, 5) \), \( Z = 26 \)
- At \( (0, 8) \), \( Z = 32 \)

### Step 4: Determine the Maximum Value
The maximum value of \( Z \) occurs at the corner point \( (0, 8) \) where \( Z = 32 \).

### Final Answer
The maximum value of \( Z \) occurs at the point \( (0, 8) \).

---