Question

Find the maximum value of \( Z = 3x + 4y \) under the constraints \[ x + y \leq 4, x \geq 0, y \geq 0. \]

Hint:

To maximize a linear objective function under constraints, evaluate the function at the vertices of the feasible region and select the maximum value.

Answer 1

We are asked to maximize the objective function \( Z = 3x + 4y \) subject to the following constraints: \[ x + y \leq 4, x \geq 0, y \geq 0. \]

Step 1: Plot the constraints.
The constraint \( x + y \leq 4 \) represents a region below the line \( x + y = 4 \). The inequalities \( x \geq 0 \) and \( y \geq 0 \) restrict the solution to the first quadrant. The feasible region is a triangle with vertices at \( (0, 0) \), \( (4, 0) \), and \( (0, 4) \).

Step 2: Evaluate \( Z \) at the vertices.
We now evaluate \( Z = 3x + 4y \) at the three vertices of the feasible region: - At \( (0, 0) \), \( Z = 3(0) + 4(0) = 0 \). - At \( (4, 0) \), \( Z = 3(4) + 4(0) = 12 \). - At \( (0, 4) \), \( Z = 3(0) + 4(4) = 16 \).

Step 3: Find the maximum value.
The maximum value of \( Z \) occurs at \( (0, 4) \), where \( Z = 16 \). Conclusion: The maximum value of \( Z = 3x + 4y \) is \( \boxed{16} \), which occurs at the point \( (0, 4) \).