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

Abbott, P. "In and Out: Acton's Railroad Problem." Mathematica J.7, 448-450, 2000.Acton, F. S. Numerical
Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., 1990.