Question

The maximum value of \(Z = 10x + 25y\) subject to \[ 0 \le x \le 3,\; 0 \le y \le 3,\; x + y \le 5,\; x \ge 0,\; y \ge 0 \] is

• A. 110
• B. 100
• C. 120
• D. 95

Hint:

In linear programming, the optimum value of the objective function always occurs at a corner point of the feasible region.

Answer 1

The Correct Option is D

Step 1: Identify the feasible region.
The constraints define a bounded polygon in the first quadrant given by \[ 0 \le x \le 3,\quad 0 \le y \le 3,\quad x+y \le 5 \] Step 2: Find corner points of the region.
The feasible corner points are \[ (0,0),\ (3,0),\ (3,2),\ (2,3),\ (0,3) \] Step 3: Evaluate \(Z\) at each corner point.
\[ Z(0,0)=0 \] \[ Z(3,0)=30 \] \[ Z(3,2)=10(3)+25(2)=30+50=80 \] \[ Z(2,3)=10(2)+25(3)=20+75=95 \] \[ Z(0,3)=75 \] Step 4: Determine the maximum value.
The maximum value of \(Z\) is \(95\), attained at \((2,3)\).