Question

The difference between the maximum value and minimum value of the objective function \(z = 3x+5y\) of a linear programming problem subject to constraints \(5x+10y\leq 50\), \(x+y\geq 1\), \(y\leq 4\) and \(x\geq 0,y\geq 0\) is \(3λ\). Then the value of \(λ\) is

• A. \(3\).
• B. \(6\).
• C. \(9\).
• D. \(27\).

Hint:

Find the corner points of the feasible region and compare z at each one.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In a linear programming problem, the maximum and minimum of the objective function occur at corner points of the feasible region.

Step 2: Draw the region:
The constraints are \(x+2y\leq10\), \(x+y\geq1\), \(y\leq4\), \(x,y\geq0\). The boundaries meet at \((1,0)\), \((10,0)\), \((2,4)\), \((0,4)\) and \((0,1)\). For example, \(y=4\) meets \(x+2y=10\) at \(x=2\).

Step 3: Evaluate z = 3x + 5y:
At \((1,0)\): 3. At \((10,0)\): 30. At \((2,4)\): 26. At \((0,4)\): 20. At \((0,1)\): 5.

Step 4: Find the difference:
Maximum is \(30\) and minimum is \(3\). The difference is \(27\). Given \(3\lambda=27\), we get \(\lambda=9\).

Step 5: Why the other options are wrong:
Option (D) 27 is the difference itself, not \(\lambda\), so we still must divide by 3. Options (A) 3 and (B) 6 would need a difference of 9 or 18, which no pair of corner values gives.

Final Answer:
So \(\lambda=9\), which is option (C). \[ \boxed{9} \]