In the graph for Linear Programming Problem, with objective function \( Z=3x+4y \) having shaded region as feasible region then, the maximum value of Z is, (Lines \( \mathrm{L}_1 \) and \( \mathrm{L}_2 \) are parallel)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 76
The maximum of a linear function over a bounded region occurs at a corner point.
The line \( \mathrm{L}_2 \) passes through (0, 0) and (12, 6), so its slope is \( \frac{1}{2} \).
The line \( \mathrm{L}_1 \) is parallel to it and passes through (0, 4), so its equation is \( y=\frac{x}{2}+4 \). At \( x=12 \) it gives \( y=10 \), so the fourth corner is (12, 10).
\( Z(0,0)=0 \)
\( Z(0,4)=16 \)
\( Z(12,6)=36+24=60 \)
\( Z(12,10)=36+40=76 \)
The maximum value of Z is 76.