Question

In the following figure, the shaded region represents the system of constraints:

• A. \(2x+y\leq 12,x+2y\leq 12,x+1.25y\geq 5,x\leq 0,y\geq 0\)
• B. \(2x+y\leq 12,x+2y\leq 12,x+1.25y\geq 5,x\geq 0,y\leq 0\)
• C. \(2x+y\leq 12,x+2y\leq 12,x+1.25y\leq 5,x\geq 0,y\geq 0\)
• D. \(2x+y\leq 12,x+2y\leq 12,x+1.25y\geq 5,x\geq 0,y\geq 0\)

Hint:

Find the corner points of the feasible region and evaluate Z at each.

Answer 1

The Correct Option is D

Step 1: Reason with the direction of increase:
$Z = 3x + 5y$ increases as x and y increase. So the smallest value is at the corner nearest the origin and the largest at a corner far from it.

Step 2: Extremes:
The smallest Z is at the corner nearest the origin, $(5, 5)$, with $Z = 40$. The largest is at the farthest corner on the line $x + 3y = 60$ with larger x, which is $(15, 15)$, $Z = 120$, compared with 100 at $(0, 20)$. Difference $120 - 40 = 80$ (D).

Final Answer:
80. \[ \boxed{80} \]