If you are in a hot-air balloon at a height of 100 meters, you can expect to see ____ km. Round your answer to the nearest kilometer and enter only the number.

Answer 1

if the earth's radius is r, then at height h, the distance d you can see is

r^2 + d^2 = (r+h)^2

However, that is just the distance from the balloon to the horizon. The actual distance seen along the earth is the part of the circumference subtended by the angle t, where

cos(t) = r/(r+h)

the distance measured along the surface of the earth is then

d = r*arccos(r/(r+h))

don't forget to express r and h in the same units, and use many significant digits, as h is very small compared to r.