A railroad one mile long is anchored at both ends. On a hot day it expands one foot and buckles. Approximately how high off the ground is it at its midpoint?

[Editor's Notes: (1) You can assume the buckled rail has the shape of an arc of a circle. Books on Strength of Materials would assume a parabolic shape. (2) The problem asks for an "approximate" answer; how "approximate" is your answer? Try to get upper and lower bounds for the true height.]

Hint 1

Draw a picture. Include the radii of the arc.

Hint 2

Here's a picture:
railroad track
We've used letters here for all the measurements; we already know that 2c = 5281 feet and 2d = 5280 feet. If we knew the length k we could calculate the height h using the Pythagorean theorem, but we don't know k. Try to get an upper bound for k (this is easy), then get an upper bound for h. Then try to get a lower bound for k (this is a lot harder) and use that to get a lower bound for h.

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