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Question #41794

Difficulty: 8/10 | 🟢 Free

The Question

Consider a Linear Programming Problem: Maximize $Z = c_1 x + c_2 y$ subject to the constraints: $2x + y \le 20$ $x + 2y \le 22$ $x \ge 0, y \ge 0$ where the cost coefficients are given by: $c_1 = \lim_{n \to \infty} \sum_{k=1}^n \ln\left(1 + \frac{4k}{n^2}\right)$ $c_2 = \lim_{n \to \infty} \sum_{k=1}^n n \left( e^{\frac{6k}{n^3}} - 1 \right)$ The maximum value of $Z$ is:

Options

A.
33
B.
36
C.
48
D.
66

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Word Problem Modeling|General vs Particular Solutions|Sum to n terms vs. General Term|Infinite Limit Behavior

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between Riemann sums and limits of sums requiring asymptotic expansion (Taylor series).

A common trap is attempting to force the limits into the standard definite integral form $\lim_{n \to \infty} \frac{1}{n} \sum f(k/n)$ , which fails here due to the $n^2$ and $n^3$ denominators.

Students must recognize that for large $n$ , the arguments of the logarithmic and exponential functions approach zero, allowing first-order Taylor approximations.

Failing to rigorously evaluate these limits leads to incorrect objective function coefficients, derailing the subsequent Linear Programming optimization.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Word Problem Modeling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free