The maximum value of z = 4x +2y subject to the constraints `2x+3yle18,x+yge10,x,yge0` is

Options

A. 36
B. 40
C. 20
D. None of these

Answer

To solve the problem of maximizing \( z = 4x + 2y \) subject to the constraints \( 2x + 3y \leq 18 \), \( x + y \geq 10 \), and \( x, y \geq 0 \), we will follow these steps:

### Step 1: Identify the Constraints
The constraints given are:
1. \( 2x + 3y \leq 18 \)
2. \( x + y \geq 10 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

### Step 2: Convert Inequalities to Equations
To find the boundary lines of the constraints, we convert the inequalities into equations:
1. \( 2x + 3y = 18 \)
2. \( x + y = 10 \)

### Step 3: Find Intercepts of the Constraints
For the first equation \( 2x + 3y = 18 \):
- When \( x = 0 \): \( 3y = 18 \) → \( y = 6 \) (y-intercept)
- When \( y = 0 \): \( 2x = 18 \) → \( x = 9 \) (x-intercept)

For the second equation \( x + y = 10 \):
- When \( x = 0 \): \( y = 10 \) (y-intercept)
- When \( y = 0 \): \( x = 10 \) (x-intercept)

### Step 4: Plot the Lines on a Graph
Plot the intercepts on a graph:
- The line \( 2x + 3y = 18 \) passes through points \( (9, 0) \) and \( (0, 6) \).
- The line \( x + y = 10 \) passes through points \( (10, 0) \) and \( (0, 10) \).

### Step 5: Determine the Feasible Region
- For \( 2x + 3y \leq 18 \), shade below the line.
- For \( x + y \geq 10 \), shade above the line.
- Also, consider the non-negativity constraints \( x \geq 0 \) and \( y \geq 0 \).

### Step 6: Identify the Vertices of the Feasible Region
Find the intersection points of the lines:
1. Solve \( 2x + 3y = 18 \) and \( x + y = 10 \):
- From \( x + y = 10 \), we can express \( y = 10 - x \).
- Substitute into \( 2x + 3(10 - x) = 18 \):
\[ 2x + 30 - 3x = 18 \implies -x + 30 = 18 \implies x = 12 \]
- Substitute \( x = 12 \) back into \( y = 10 - x \):
\[ y = 10 - 12 = -2 \quad (\text{Not feasible since } y \geq 0) \]

2. Check the intercepts:
- \( (9, 0) \) and \( (0, 6) \) from the first constraint.
- \( (10, 0) \) and \( (0, 10) \) from the second constraint.

### Step 7: Evaluate the Objective Function at the Vertices
Evaluate \( z = 4x + 2y \) at the feasible points:
- At \( (9, 0) \): \( z = 4(9) + 2(0) = 36 \)
- At \( (0, 6) \): \( z = 4(0) + 2(6) = 12 \)
- At \( (10, 0) \): \( z = 4(10) + 2(0) = 40 \)
- At \( (0, 10) \): \( z = 4(0) + 2(10) = 20 \)

### Step 8: Determine the Maximum Value
The maximum value of \( z \) occurs at \( (10, 0) \):
\[ \text{Maximum value of } z = 40 \]

### Conclusion
The maximum value of \( z = 4x + 2y \) subject to the given constraints is **40**.