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 determine the maximum value of the objective function \( z = 3x + 4y \) using linear programming, we consider the following constraints:

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

The feasible region is graphed by plotting these constraints. The line \( x + y = 4 \) intersects the axes at \( (4,0) \) and \( (0,4) \). The conditions \( x \geq 0 \) and \( y \geq 0 \) restrict the region to the first quadrant.

Consequently, the feasible region is a triangle bounded by the vertices \( (0,0) \), \( (4,0) \), and \( (0,4) \).

We then evaluate the objective function \( z = 3x + 4y \) at each vertex to find the maximum value:

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

The highest value obtained for \( z \) at these vertices is \( 16 \), which occurs at the point \( (0,4) \).

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