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Question #40553

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the feasible region in the $xy$ -plane defined by the inequalities: $\frac{x + 2y - 8}{y - 3} \le 2, \quad 0 \le x \le 6, \quad 0 \le y \le 6$ If $[ \cdot ]$ denotes the greatest integer function, find the maximum value of the objective function $Z = 7x + 3[y]$ on the region $S$ .

Options

A.
32
B.
48
C.
51
D.
60

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Graphical Feasible Region|Global Extrema on Intervals|Greatest Integer Function Traps|Sign Flipping Rule

Why This Question Is Tricky

Common Mistakes

This question targets the 'Sign Flipping Rule' and 'GIF Trap' in optimization.

Students commonly cross-multiply the inequality without considering the sign of $(y-3)$ , implicitly assuming $y > 3$ .

This leads them to erroneously conclude $x \le 2$ and find a false maximum of 32.

Furthermore, for those who correctly identify the region $y < 3$ , the Greatest Integer Function creates a trap at the open boundary.

Evaluating at the vertex $(6,3)$ yields 51, but since $y=3$ is strictly excluded from the domain, the supremum of $[y]$ is 2, making the true global maximum 48.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Graphical Feasible Region
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free