Question

If $Z = 10x + 25y$ subject to $0 \le x \le 3$, $0 \le y \le 3$, $x + y \le 5$, $x \ge 0$, $y \ge 0$, then $Z$ is maximum at the point

• A. $(2,4)$
• B. $(1,6)$
• C. $(2,3)$
• D. $(4,3)$

Hint:

In linear programming problems, the maximum or minimum value of the objective function always occurs at one of the corner points of the feasible region.

Answer 1

The Correct Option is C

Step 1: Understanding the problem.
The given problem is a linear programming problem where the objective function is \[ Z = 10x + 25y \] subject to the constraints \[ 0 \le x \le 3,\quad 0 \le y \le 3,\quad x + y \le 5. \] To find the maximum value of $Z$, we evaluate the objective function at all feasible corner points of the region.
Step 2: Identifying feasible corner points.
The constraints form a bounded region in the first quadrant. The feasible corner points are: \[ (0,0),\ (3,0),\ (0,3),\ (2,3),\ (3,2). \]
Step 3: Evaluating the objective function at each corner point.
\[ Z(0,0) = 10(0) + 25(0) = 0 \] \[ Z(3,0) = 10(3) + 25(0) = 30 \] \[ Z(0,3) = 10(0) + 25(3) = 75 \] \[ Z(2,3) = 10(2) + 25(3) = 20 + 75 = 95 \] \[ Z(3,2) = 10(3) + 25(2) = 30 + 50 = 80 \]
Step 4: Conclusion.
The maximum value of $Z$ is $95$, which occurs at the point $(2,3)$.