Long Railway Ties

A friend of mine gave me this puzzle. Assuming that you had perfectly straight and flat railroad ties that were laid end to end. Each tie is one mile long. But the manufacturer made a set of ties one foot too long. If we place the long ties in between a set of ties that have already been laid one mile apart, the the long ties will bulge vertically. How high will this bulge be?

I guessed a very low figure first. Then I solved the problem using pythagorean theorem and was surprised when I calculated the height (over 51 ft) was much higher than I would have guessed. But the long railroad tie does not form two right triangles.

So, what is the curve that the 5281-ft tie forms when fitted in a mile-long space? And what is the actual height of the bulge that the tie forms?