Limits, Continuity & Differentiability
Limits of Sequences
Grade 12
Question:
<p>A circular disk of unit radius is filled with a number of smaller circular disks arranged in the form of hexagon. Let \(A_n\) denotes a stack of disks arranged in the shape of a hexagon having \(n\) disks on a side. If \(A\) be the area of large disk, \(S_n\) be the number of disks in \(A_n\) configuration and \(r_n\) be the radius of each disk in \(A_n\) configuration, then find \(\lim_{n \to \infty} n r_n\)</p>
<p>(a) \(\frac{1}{3}\)</p>
<p>(b) \(\frac{1}{2}\)</p>
<p>(c) \(\frac{1}{4}\)</p>
<p>(d) \(\frac{1}{11}\)</p>
Step-by-Step Solution
Key Concept: The radius of disks in the hexagonal packing decreases inversely with $n$, so $r_n \sim \frac{c}{n}$ for some constant.
<p>Since the large disk has unit radius and the smaller disks are packed in a hexagonal arrangement, the radius of each disk in configuration an is approximately $r_n \approx \frac{1}{2n}$. Therefore, $$\lim_{n \to \infty} n r_n = \lim_{n \to \infty} n \cdot \frac{1}{2n} = \frac{1}{2}$$</p>
Correct Answer: b