Question

The maximum value of Z = 3x + 4y subject to constraint x + y ≤6, x, y ≥ 0 is:

• A. 18
• B. 20
• C. 22
• D. 24

Answer 1

The Correct Option is D

To find the maximum value of \( Z = 3x + 4y \) subject to the constraint \( x + y \leq 6 \) and \( x, y \geq 0 \), we need to determine the feasible region and evaluate the objective function at the vertices of this region. The constraint can be rewritten as \( y \leq 6 - x \).

1. Graphical Representation:

Let's plot the line \( y = 6 - x \) and consider the constraints:

• \( x \geq 0 \)
• \( y \geq 0 \)
• \( x + y \leq 6 \)

The feasible region is a triangle with vertices at \( (0,0) \), \( (6,0) \), and \( (0,6) \).

2. Evaluate the Objective Function at Each Vertex:

\(Z = 3x + 4y\)	Vertex \( (x,y) \)	Value of \( Z \)
\((0,0)\)	\(Z = 3(0) + 4(0) = 0\)
\((6,0)\)	\(Z = 3(6) + 4(0) = 18\)
\((0,6)\)	\(Z = 3(0) + 4(6) = 24\)

3. Determine the Maximum Value:

The maximum value of \( Z \) is \( 24 \) at the vertex \( (0,6) \).

Therefore, the maximum value of \( Z = 3x + 4y \) under the given constraints is \( \mathbf{24} \).