Question

If the difference between the maximum and minimum values of the objective function \(z = 7x - 8y\), subject to the constraints \(x + y \le 20, y \ge 5, x, y \ge 0\) is \(5k + 200\), then the value of k is

• A. \(3\)
• B. \(4\)
• C. \(5\)
• D. \(6\)

Hint:

In linear programming, always evaluate the objective function only at the corner points of the feasible region.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Optimal values of an objective function in LPP occur at the corner points (vertices) of the feasible region.
Step 2: Key Formula or Approach:
Identify vertices from the intersection of boundaries: \(x+y=20, y=5, x=0\).
Step 3: Detailed Explanation:
Feasible region vertices:
- Intersection of \(y=5\) and Y-axis (\(x=0\)): \((0, 5)\).
- Intersection of \(x+y=20\) and \(y=5\): \(x+5=20 \implies (15, 5)\).
- Intersection of \(x+y=20\) and Y-axis (\(x=0\)): \((0, 20)\).
Calculate \(z = 7x - 8y\) at each vertex:
- At \((0, 5)\): \(z = 0 - 40 = -40\).
- At \((15, 5)\): \(z = 7(15) - 8(5) = 105 - 40 = 65\).
- At \((0, 20)\): \(z = 0 - 160 = -160\).
Max value \(= 65\), Min value \(= -160\).
Difference \(= 65 - (-160) = 225\).
Given Difference \(= 5k + 200\).
\[ 5k + 200 = 225 \implies 5k = 25 \implies k = 5 \] Step 4: Final Answer:
The value of \(k\) is \(5\).