Suppose a railroad rail is 3 kilometers and it expands on a hot day by 12 centimeters in length. approximately how many meters would the center of the rail rise above the ground

For small angles, x, in radians,
tan(x) ≈ sin(x) = x

Use this to create an approximate diagram for the problem as shown in the figure below.
The actual deflected shape is parabolic, but it is approximated by an isosceles triangle with small equal angles.

The extend length is 3 km + 12 cm, which is 
3000 + 0.12 = 3000.12 m

Half of the extended length is 1500.06 m

Let h+  the rise of the center of the rail above ground.
From the Pythagorean theorem, obtain
h² + 1500² = 1500.6²
h² = 1500.06² - 1500² = 180.004
h = 13.4 m, which is about 0.9% of the original length.

  13.4 m