Maximum value of `Z = 3x + 4y` subject to the constraints `x + y le 4, x ge 0, y ge 0` is 16.

Answer

To solve the problem of maximizing \( Z = 3x + 4y \) 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
We have the following constraints:
1. \( x + y \leq 4 \)
2. \( x \geq 0 \)
3. \( y \geq 0 \)

### Step 2: Graph the Constraints
To graph the constraint \( x + y \leq 4 \):
- Convert it to the equation \( x + y = 4 \).
- Find the intercepts:
- When \( x = 0 \), \( y = 4 \) (point B: \( (0, 4) \)).
- When \( y = 0 \), \( x = 4 \) (point A: \( (4, 0) \)).

Now, plot these points on a graph and draw the line connecting them. The area below this line represents the feasible region defined by \( x + y \leq 4 \).

### Step 3: Identify the Feasible Region
The feasible region is bounded by:
- The line \( x + y = 4 \)
- The x-axis (where \( y = 0 \))
- The y-axis (where \( x = 0 \))

The vertices of the feasible region are:
- Point A: \( (4, 0) \)
- Point B: \( (0, 4) \)
- Point C: \( (0, 0) \)

### Step 4: Evaluate the Objective Function at Each Vertex
Now we will evaluate \( Z = 3x + 4y \) at each vertex:
1. At point A \( (4, 0) \):
\[ Z = 3(4) + 4(0) = 12 \]
2. At point B \( (0, 4) \):
\[ Z = 3(0) + 4(4) = 16 \]
3. At point C \( (0, 0) \):
\[ Z = 3(0) + 4(0) = 0 \]

### Step 5: Determine the Maximum Value
From the evaluations:
- At \( (4, 0) \), \( Z = 12 \)
- At \( (0, 4) \), \( Z = 16 \)
- At \( (0, 0) \), \( Z = 0 \)

The maximum value of \( Z \) is \( 16 \) at the point \( (0, 4) \).

### Conclusion
Thus, the maximum value of \( Z = 3x + 4y \) subject to the given constraints is \( 16 \).

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