The distance l from a point at a height h above the Earth's surface to the horizon can be approximated using Pythagoras' theorem by the expression: (Please see the attachment below)
(a) Find an expression which serves as a linear approximation for l at h=1000 m.
(b) Give two assumptions you think have been made in deriving the expression above.
Please see the attachment below for the complete solution.
The above equation can be written in a more useful form as:
l = (h^2+2rh)^1/2
Taking the derivative of l with respect to h gives:
dl/dh = ...
This shows how to perform linearization of a function related to the Pythagorean theorem.