The maximum value of z = 3x + 2y subject to `x + 2y ge 2 , x + 2y lt 8, x, y ge 0` is :

Options

A. 32
B. 24
C. 40
D. None of these

Answer

To solve the problem of maximizing \( z = 3x + 2y \) subject to the constraints \( x + 2y \geq 2 \), \( x + 2y < 8 \), and \( x, y \geq 0 \), we can follow these steps:

### Step 1: Identify the Constraints
The constraints are:
1. \( x + 2y \geq 2 \)
2. \( x + 2y < 8 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

### Step 2: Convert Inequalities to Equations
To find the boundary lines, we convert the inequalities into equations:
1. \( x + 2y = 2 \)
2. \( x + 2y = 8 \)

### Step 3: Find Intercepts of the Lines
For the line \( x + 2y = 2 \):
- Set \( x = 0 \): \( 2y = 2 \) → \( y = 1 \) → Point (0, 1)
- Set \( y = 0 \): \( x = 2 \) → Point (2, 0)

For the line \( x + 2y = 8 \):
- Set \( x = 0 \): \( 2y = 8 \) → \( y = 4 \) → Point (0, 4)
- Set \( y = 0 \): \( x = 8 \) → Point (8, 0)

### Step 4: Plot the Lines
Now, we plot the points (0, 1), (2, 0), (0, 4), and (8, 0) on the coordinate system.

### Step 5: Identify the Feasible Region
- The line \( x + 2y = 2 \) is a boundary for \( x + 2y \geq 2 \), which means the region above this line is included.
- The line \( x + 2y = 8 \) is a boundary for \( x + 2y < 8 \), which means the region below this line is included.
- Since \( x \geq 0 \) and \( y \geq 0 \), we are only interested in the first quadrant.

### Step 6: Determine the Vertices of the Feasible Region
The feasible region is bounded by the points:
- (0, 1)
- (2, 0)
- (0, 4)
- (8, 0)

### Step 7: Evaluate the Objective Function at Each Vertex
Now, we evaluate \( z = 3x + 2y \) at each vertex:
1. At (0, 1): \( z = 3(0) + 2(1) = 2 \)
2. At (2, 0): \( z = 3(2) + 2(0) = 6 \)
3. At (0, 4): \( z = 3(0) + 2(4) = 8 \)
4. At (8, 0): \( z = 3(8) + 2(0) = 24 \)

### Step 8: Identify the Maximum Value
The maximum value of \( z \) occurs at the point (8, 0):
- Maximum \( z = 24 \)

### Conclusion
The maximum value of \( z = 3x + 2y \) subject to the given constraints is **24** at the point **(8, 0)**.

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