The maximum value of the objective function
Z=3x+4y
subject to the constraints ` x+y le 4, x ge 0 , y le o ` is

Options

A. 16
B. 18
C. 20
D. 25

Explanation

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

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

### Step 2: Graph the constraints
To graph the constraints, we first convert the inequality \( x + y \leq 4 \) into an equation:
- The line \( x + y = 4 \) can be rewritten as \( y = 4 - x \).

Next, we find the intercepts:
- When \( x = 0 \), \( y = 4 \) (point \( (0, 4) \)).
- When \( y = 0 \), \( x = 4 \) (point \( (4, 0) \)).

Now, we plot these points on a coordinate system and draw the line connecting them. The feasible region is below this line and in the first quadrant (since \( x \geq 0 \) and \( y \geq 0 \)).

### Step 3: Identify the vertices of the feasible region
The vertices of the feasible region are:
1. \( (0, 0) \)
2. \( (0, 4) \)
3. \( (4, 0) \)

### Step 4: Evaluate the objective function at each vertex
Now, we will evaluate the objective function \( Z = 3x + 4y \) at each of the vertices:

1. At \( (0, 0) \):
\[ Z = 3(0) + 4(0) = 0 \]

2. At \( (0, 4) \):
\[ Z = 3(0) + 4(4) = 16 \]

3. At \( (4, 0) \):
\[ Z = 3(4) + 4(0) = 12 \]

### Step 5: Determine the maximum value
Now, we compare the values of \( Z \) at each vertex:
- At \( (0, 0) \), \( Z = 0 \)
- At \( (0, 4) \), \( Z = 16 \)
- At \( (4, 0) \), \( Z = 12 \)

The maximum value of \( Z \) occurs at the vertex \( (0, 4) \) where \( Z = 16 \).

### Final Answer
The maximum value of the objective function \( Z = 3x + 4y \) subject to the given constraints is **16**.

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