Question

The minimum value of \( Z = 5x + 8y \) subject to \[ x + y \geq 5, \quad 0 \leq x \leq 4, \quad y \geq 2, \quad x \geq 0, \quad y \geq 0 \] is

• A. 40
• B. 36
• C. 31
• D. 20

Hint:

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

Answer 1

The Correct Option is C

Step 1: Identifying the feasible region.
The constraints define a feasible region where the objective function \( Z = 5x + 8y \) must be minimized. The corners of the feasible region are key points where we can evaluate the objective function.

Step 2: Evaluating the objective function.
We evaluate \( Z \) at the corners of the feasible region. The points of interest are the intersections of the lines defined by the constraints. After calculation, the minimum value of \( Z \) is found to be 31.

Step 3: Conclusion.
Thus, the minimum value of \( Z \) is 31, which makes option (C) the correct answer.