Question

Minimize \[ Z=2x+3y \] subject to \[ x+y\ge5, \qquad x,y\ge0. \]

• A. \(10\)
• B. \(15\)
• C. \(0\)
• D. \(12\)

Hint:

For minimization problems, always select the smallest value of the objective function among the feasible corner points.

Answer 1

The Correct Option is A

Concept: In Linear Programming Problems, the optimum value of the objective function occurs at one of the corner points of the feasible region. The objective function is \[ Z=2x+3y \] and the constraint is \[ x+y\ge5 \] with \[ x\ge0,\qquad y\ge0. \]

Step 1: Determine the feasible region The line \[ x+y=5 \] cuts the coordinate axes at \[ (5,0) \] and \[ (0,5). \] Since \[ x+y\ge5, \] the feasible region lies on or above the line. The corner points on the boundary are \[ (5,0) \] and \[ (0,5). \]

Step 2: Evaluate the objective function at corner points At \[ (5,0) \] \[ Z=2(5)+3(0) \] \[ =10 \] At \[ (0,5) \] \[ Z=2(0)+3(5) \] \[ =15 \]

Step 3: Compare the values \[ Z(5,0)=10 \] \[ Z(0,5)=15 \] The smaller value is \[ 10 \] Final Answer: \[ \boxed{10} \]