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Question #41693

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the feasible region defined by the system of inequalities: $x \ge 0, \quad y \ge 0$ $x + y \le 14$ $24 \le 3x + 4y \le 24 + 5d$ where $d > 0$ is the perpendicular distance between the parallel lines $3x + 4y = 24$ and $3x + 4y = 24 + 5d$ . If the maximum value of the objective function $Z = 4x + 5y$ on the region $S$ is $62$ , find the area of the region $S$ .

Options

A.
66
B.
72
C.
74
D.
63

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

Corner Point Method Precision|Geometry Constraint Modeling|Slope Sensitivity|Distance Between Parallel Lines

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to use 'Slope Sensitivity' to identify the optimal corner point in a parameterized Linear Programming Problem.

The trap lies in assuming the maximum occurs at the axis intercepts of the moving parallel line $3x+4y=24+5d$ .

Because the slope of the objective function ( $-4/5$ ) lies strictly between the slopes of the two intersecting constraints ( $-1$ and $-3/4$ ), the optimal solution is locked at their intersection point $V(32-5d, 5d-18)$ as the line translates.

Failing to compare slopes leads to incorrect locus tracking, resulting in a wrong value for $d$ (often $d=4.5$ or $d=5.12$ ) and consequently an incorrect area calculation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Corner Point Method Precision
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free