Question

Maximize \( z = 3x + 4y \) subject to \( x + y \leq 4 \), \( x \geq 0 \), \( y \geq 0 \). What is the maximum value of \( z \)?

• A. \( 12 \)
• B. \( 16 \)
• C. \( 14 \)
• D. 10

Hint:

In LPP, evaluate the objective function at the vertices of the feasible region to find the maximum.

Answer 1

The Correct Option is B

To find the maximum value of \( z = 3x + 4y \), we use the method of linear programming. The objective function is subject to the constraints:

\( x + y \leq 4 \), \( x \geq 0 \), \( y \geq 0 \).

First, we identify the feasible region by plotting the constraints on a graph:

• \( x + y \leq 4 \) is a line with intercepts at \( (4,0) \) and \( (0,4) \).
• The inequalities \( x \geq 0 \) and \( y \geq 0 \) indicate the feasible region is in the first quadrant.

The feasible region is a triangle with vertices at \( (0,0) \), \( (4,0) \), and \( (0,4) \).

We evaluate the objective function \( z = 3x + 4y \) at each vertex:

Vertex	\( z = 3x + 4y \)
\((0,0)\)	\(0\)
\((4,0)\)	\(3(4) + 4(0) = 12\)
\((0,4)\)	\(3(0) + 4(4) = 16\)

The maximum value of \( z \) at these vertices is \( 16 \), occurring at \( (0,4) \).

Thus, the maximum value of \( z \) is \(\boxed{16}\).

Answer 2

To maximize the objective function \( z = 3x + 4y \) subject to the constraints \( x + y \leq 4 \), \( x \geq 0 \), and \( y \geq 0 \), we follow these steps:

1. Define the Feasible Region: The constraints are:

• \( x + y \leq 4 \)
• \( x \geq 0 \)
• \( y \geq 0 \)

These constraints describe a triangle with vertices at: (0,0), (4,0), and (0,4).

2. Evaluate the Objective Function at Each Vertex:

Vertex	Objective Function \( z = 3x + 4y \)
(0,0)	\( z = 3(0) + 4(0) = 0 \)
(4,0)	\( z = 3(4) + 4(0) = 12 \)
(0,4)	\( z = 3(0) + 4(4) = 16 \)

3. Determine the Maximum Value: Among the calculated values of \( z \), the highest is 16, which occurs at the point (0,4).

Therefore, the maximum value of \( z \) is 16.