Railroad Track Problem

Given a straight segment of track of length , add a small segment so that the track bows into a circular arc. Find the maximum displacement of the bowed track. The Pythagorean theorem gives

But is simply , so

Solving (1) and (2) for gives

Expressing the length of the arc in terms of the central angle,

But is given by

so plugging in gives

This is a transcendental equation that cannot be solved exactly with a closed-form solution for , but for ,

Therefore,

Keeping only terms to order ,

so

and

If we take and 1 foot, then feet. Solving equation (◇) numerically, we find that the true answer is feet.