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Question #41766

Difficulty: 8/10 | 🟢 Free

The Question

Let $R$ be the feasible region in $\mathbb{R}^3$ defined by the following system of inequalities: $x \ge 0, y \ge 0, z \ge 0$ $2x + y + 3z \le 12$ $x + 3y + 2z \le 12$ $3x + 4y + 5z \ge 24$ If $L$ is the maximum distance between any two points in $R$ , find the value of $49 L^2$ .

Options

A.
1728
B.
576
C.
1152
D.
864

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

Feasible Region Constancy|Coplanarity Condition|Linear Independence vs Dependence

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to identify linear dependence among constraint normals in a 3D optimization problem.

The trap lies in assuming the inequalities define a standard 3D polyhedron with non-zero volume.

Students who blindly apply vertex-finding algorithms or the simplex method will struggle, as the standard intersections of planes do not yield feasible points.

By recognizing that the sum of the first two constraints provides an upper bound that perfectly contradicts the lower bound of the third constraint, the student deduces that the region collapses into a 1D line segment.

The problem then elegantly transitions into 3D coordinate geometry, requiring the parameterization of this segment to find its endpoints and length.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Feasible Region Constancy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free