Imagine you would have to put a rope around the moon. Since the moon is 1,738,000 metres in diameter, this is a hard task.
Finally you have managed to get the rope around the moon but... it is one meter short.
You decide to dig a groove all around the moon, so that the shorter rope suffices. How deep must this groove be?
(Assume the Moon to be a perfect sphere.)
groove depth of 1/(2*pi) meters ~ 0.159 meters
Current circumference = 1738000*pi, which means the rope is 1738000*pi-1. This will be the new circumference as well. The new diameter will be 1738000 - 2 * g (g = groove depth, x2 because one on each side). So new circumference = (1738000 - 2g)*pi. So 1738000*pi - 1 = (1738000 - 2g)*pi. Work this down, I get a groove depth of 1/(2*pi)
Solution
