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: Feasible region:
Constraints: \(x + 3y \le 60\), \(x + y \ge 10\), \(x \le y\), \(x, y \ge 0\).

Step 2: Corner points:
Intersections of boundary lines: \(x = y\) with \(x + y = 10\) gives \((5, 5)\). \(x = y\) with \(x + 3y = 60\) gives \((15, 15)\). The y axis meets \(x + y = 10\) at \((0, 10)\) and \(x + 3y = 60\) at \((0, 20)\).

Step 3: Evaluate Z:
\(Z(5,5) = 40\), \(Z(15,15) = 120\), \(Z(0,10) = 50\), \(Z(0,20) = 100\).
Maximum = 120 and minimum = 40. Difference = 80.

Final Answer:
The difference is 80, option (D). \[ \boxed{80} \]