Question

suppose a railroad rail is 1 kilometer long and it expands on a hot day by 10 centimeters in length. how high would the center of the rail rise above the ground?

Answer 1

First:
railways are designed to NOT do this otherwise there would be crashes every jot day

Second:
It depends on how you model the shape of the curve that it makes.

Since the effects are VERY small I suggest you use 2 triangles for simplicity:



Use Pythagoras to estimate h