Inspired by toilet paper

Geometry Level 3

Referring to the figure above, suppose \(A_1\) is area of first white circle, then \(A_2\) is area of pink annulus, \(A_3\) is area of blue annulus and like wise \(A_n\) would be the annulus bounded by \(n^\text{th}\) and \((n-1)^\text{th}\) circle. These \(n\) concentric circles are drawn such that the area bounded by two concentric circles remains constant, that is \(A_1=A_2=A_3= \cdots = A_n\).

Define \(r_n\) to be the radius of \(n^\text{th}\) circle.Evaluate \(\dfrac{r_{100}}{r_1}\) correct 2 decimal places.


Inspiration.

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