Posted 2 Mar 2013 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? 0 Share this post Link to post Share on other sites

0 Posted 2 Mar 2013 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 x, and so x is small. The Maclaurin series for tan x is: x + x^{3}/3 + ... Hence x + x^{3}/3 x + d/2r, and so x^{3} 3d/2r. 0 Share this post Link to post Share on other sites

Posted

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+kwhere

kis some constant, andxis expressed in radians.You can't get a general solution

x=x(k), but you can solve the equation iteratively forxfor any particular value ofk.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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