Question

Maximize \[ Z=3x+4y \] subject to \[ x+y\le10,\qquad x,y\ge0 \]

• A. \((10,0)\)
• B. \((0,10)\)
• C. \((5,5)\)
• D. \((0,0)\)

Hint:

In LPP, always test all corner points because optimum values occur only at vertices.

Answer 1

The Correct Option is B

Concept: In Linear Programming Problems (LPP), the maximum or minimum value of the objective function occurs at the corner points of the feasible region.

Step 1: Identify constraints Given: \[ x+y\le10 \] and \[ x\ge0,\qquad y\ge0 \] These inequalities represent the feasible region in the first quadrant.

Step 2: Find corner points The line: \[ x+y=10 \] cuts the axes at: \[ (10,0)\quad \text{and}\quad (0,10) \] Including the origin, corner points are: \[ (0,0),\ (10,0),\ (0,10) \]

Step 3: Evaluate objective function Objective function: \[ Z=3x+4y \] At \((0,0)\): \[ Z=3(0)+4(0)=0 \] At \((10,0)\): \[ Z=3(10)+4(0)=30 \] At \((0,10)\): \[ Z=3(0)+4(10)=40 \]

Step 4: Choose maximum value Largest value is: \[ 40 \] obtained at: \[ (0,10) \] Final Answer: \[ \boxed{(0,10)} \]