The maximum value of Z = 4x + 2y subject to the constraints `2x+3y le 18,x+y ge 10, x , y ge 0 ` is

Options

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

Explanation

To solve the linear programming 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
We have the following constraints:
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 graph the constraints, we first convert the inequalities into equations:
1. \( 2x + 3y = 18 \)
2. \( x + y = 10 \)

### Step 3: Find the Intercepts
**For the first equation \( 2x + 3y = 18 \):**
- Set \( x = 0 \):
\[ 2(0) + 3y = 18 \implies y = 6 \quad \text{(Intercept: (0, 6))} \]
- Set \( y = 0 \):
\[ 2x + 3(0) = 18 \implies x = 9 \quad \text{(Intercept: (9, 0))} \]

**For the second equation \( x + y = 10 \):**
- Set \( x = 0 \):
\[ 0 + y = 10 \implies y = 10 \quad \text{(Intercept: (0, 10))} \]
- Set \( y = 0 \):
\[ x + 0 = 10 \implies x = 10 \quad \text{(Intercept: (10, 0))} \]

### Step 4: Graph the Constraints
Now we will graph the lines:
- The line \( 2x + 3y = 18 \) passes through points (0, 6) and (9, 0).
- The line \( x + y = 10 \) passes through points (0, 10) and (10, 0).

### Step 5: Determine the Feasible Region
- For \( 2x + 3y \leq 18 \), shade the region below the line.
- For \( x + y \geq 10 \), shade the region above the line.

### Step 6: Find the Intersection Points
To find the vertices of the feasible region, we need to find the intersection of the two lines:
1. From \( 2x + 3y = 18 \)
2. From \( x + y = 10 \)

Substituting \( y = 10 - x \) into the first equation:
\[ 2x + 3(10 - x) = 18 \]
\[ 2x + 30 - 3x = 18 \]
\[ -x + 30 = 18 \implies x = 12 \]
Substituting \( x = 12 \) back into \( y = 10 - x \):
\[ y = 10 - 12 = -2 \quad \text{(Not feasible since } y \geq 0\text{)} \]

### Step 7: Check Vertices of the Feasible Region
The feasible region is determined by the intercepts and the area where the shaded regions overlap. The vertices of the feasible region are:
- (0, 10)
- (9, 0)
- (0, 6)

### Step 8: Evaluate the Objective Function at Each Vertex
Now we will evaluate \( Z = 4x + 2y \) at each vertex:
1. At (0, 10):
\[ Z = 4(0) + 2(10) = 20 \]
2. At (9, 0):
\[ Z = 4(9) + 2(0) = 36 \]
3. At (0, 6):
\[ Z = 4(0) + 2(6) = 12 \]

### Step 9: Determine the Maximum Value
The maximum value of \( Z \) occurs at the vertex (9, 0):
\[ \text{Maximum value of } Z = 36 \]

### Conclusion
The maximum value of \( Z = 4x + 2y \) subject to the given constraints is **36**.

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