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Two puzzles, same analysis

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This is not a puzzle but an observation of something that to me was surprising.

Two unrelated puzzles - the and the problem, succumb to identical analyses.

Both present an equation that does not have an analytical solution for a crucial angle, let's call it x.

The equation that presents itself is this:

tan(x) = x + k

where k is some constant, and x is expressed in radians.

You can't get a general solution x = x(k), but you can solve the equation iteratively for x for any particular value of k.

When that is done, the the desired answer turns out to be

cos(x) = answer.

I wonder whether there is a prototypical problem for which this is the best analysis?

Did this post turn out to be a puzzle after all?

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Let d be the extra length added to the belt:
Then 2a = 2rx + d.
Hence a = rx + d/2, and so a/r = x + d/2r.

We also have, tan x = a/r.
Therefore tan x = x + d/2r.

Given numerical values for d and r, this equation can be solved for x to any required degree of accuracy using the Newton-Raphson Method (or Newton's Method), enabling us to calculate h or Instead, we could pursue an approximate solution for small values of d = extra belt length.

Assume d/2r is very small, as it is for d = 1, r = 6,400,000.
Then tan x approx.gif x, and so x is small.

The Maclaurin series for tan x is: x + x3/3 + ...

Hence x + x3/3 approx.gif x + d/2r, and so x3approx.gif 3d/2r.

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