The Other Rope Around The Earth Solution 

For the problem statement:
The rope stretches from the top of the pole to the horizon. Suppose
the angle subtended by the rope is t, the radius of the Earth is R,
and the distance to the horizon is D. Then
 (1) D^2+R^2 = (R+h)^2
 (2) D = R.tan(t)
 (3) R.t + 3 = D.
Solving is tough, but we use:
and so:
 (5) R.t + 3 = R.tan(t) [ from (2) and (3) ]
 (6) t + 3/R ~ t + t^3/3 [ from (5) and (4) ]
 (7) t^3 ~ 9/R [ rearranging (6) ]
Thus t ~ 0.0112233... and we get:
(drum roll please ...)
h ~ 401 metres.
See:
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