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

Concept:
In linear programming, the maximum and minimum values of the objective function occur at the corner points of the feasible region. ip

Step 1: Find the feasible region corner points.
Constraints are: \[ x+y\le 20,\qquad y\ge 5,\qquad x\ge 0,\qquad y\ge 0 \] The relevant corner points are: \[ (0,5),\quad (15,5),\quad (0,20) \] ip

Step 2: Evaluate the objective function.
For \[ z=7x-8y \] At \((0,5)\): \[ z=7(0)-8(5)=-40 \] At \((15,5)\): \[ z=7(15)-8(5)=105-40=65 \] At \((0,20)\): \[ z=7(0)-8(20)=-160 \] ip

Step 3: Find the difference between maximum and minimum values.
Maximum value: \[ 65 \] Minimum value: \[ -160 \] Difference: \[ 65-(-160)=225 \] Given, \[ 225=5k+200 \] \[ 5k=25 \] \[ k=5 \] ip Hence, the correct answer is:
\[ \boxed{(C)\ 5} \]