In a linear programming problem, the corner points of the feasible region, determined by the system of linear inequalities are (0, 0), (5, 0), (3, 4) and (0, 5). Let \( z=5x+qy,q\gt 0 \) be the objective function. For what value of q the maximum value of z occurs at both (3, 4) and (0, 5)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 15
If the maximum occurs at both corner points, z must have the same value at them:
\( \displaystyle z(3,4)=5(3)+4q=15+4q \)
\( \displaystyle z(0,5)=5(0)+5q=5q \)
\( \displaystyle 15+4q=5q\ \Rightarrow\ q=15 \)
This common value must also be the maximum. With q = 15,
\( \displaystyle z(0,0)=0 \)
\( \displaystyle z(5,0)=25 \)
\( \displaystyle z(3,4)=15+60=75 \)
\( \displaystyle z(0,5)=75 \)
So 75 is the maximum, attained at both (3, 4) and (0, 5).