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Acton's Railroad Problem

Q: In Numerical Methods that Work [1] there is a famous "railroad track problem": 1 foot of extra track is inserted into 1 mile of  railroad track, and it bows up in a circular arc. Find the maximum height it  achieves off the ground.


How can I obtain a numerical solution to this problem and is it analytically solvable?
A: From the figure, with [Graphics:../Images/index_gr_4.gif], we obtain the following system of equations.
Here is a one-line numerical solution.
Hence the maximum height it achieves off the ground is 44.5 feet.
Generalizing the equations by writing [Graphics:../Images/index_gr_8.gif] as the length of straight track and [Graphics:../Images/index_gr_9.gif] as the length of curved track, where [Graphics:../Images/index_gr_10.gif], and eliminating the nontrigonometric variables, we find
Even this simple equation is not analytically solvable.
However, as Ronald Bruck ( points out, it is easy to determine series representations for the  answer. Since [Graphics:../Images/index_gr_16.gif] and explicitly noting that [Graphics:../Images/index_gr_17.gif] depends on [Graphics:../Images/index_gr_18.gif], we obtain the series expansion for [Graphics:../Images/index_gr_19.gif].
Using series inversion, we obtain [Graphics:../Images/index_gr_22.gif].
We can use this expansion to determine [Graphics:../Images/index_gr_25.gif] and [Graphics:../Images/index_gr_26.gif]. Since [Graphics:../Images/index_gr_27.gif], the series for [Graphics:../Images/index_gr_28.gif] is
Since [Graphics:../Images/index_gr_31.gif], the series for [Graphics:../Images/index_gr_32.gif] is
Note that the order is important here--we need to determine [Graphics:../Images/index_gr_35.gif] before we determine [Graphics:../Images/index_gr_36.gif]. We truncate these series expansions as follows.
Then, for the original problem,
Since [Graphics:../Images/index_gr_39.gif], we find
This result is in good agreement with FindRoot.

Converted by Mathematica      May 8, 2000

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