How high the moon
Imagine two ants, or an aunt and an uncle, sitting in opposite corners of a square. One of them would like to go to the other while the other just wants to sit still. The square measures 10 by 10 metres.
There is a fence, with no thickness and varying height. The height of the fence at the corners is zero and its highest point is at the middle of the fence. Actually, the height varies in such a way that if the ant goes straight to the wall, climbs it, descends on the other side, and goes straight to the opposite corner, it doesn’t matter which route the ant takes, they are all equally long.
January 28th, 2009 at 5:21 am
I presume the question is “describe the height of the fence.”
Let ? be the angle away from the diagonal traveled by the uncle (?=0 is on the diagonal, ?=45° is along the side of the square). Then the distance from the uncle’s corner to the fence is d(?) = s/(?2 cos?), where s is the length of a side.
Since all routes are equally long, the distance to the fence, d(?), and height of the fence at that location, h(?), must sum to the length of the side, so
d(?) + h(?) = s h(?) = s[1 - 1/(?2 cos?)]
It’s shaped somewhat like a parabola.
January 28th, 2009 at 5:22 am
Arrgh! Some of the “?” are theta’s, the others are square roots (as in square root of 2).