An ant is at a crossroads of a square grid of highways spaced 1 meter apart.
The ant can walk 1 meter in 15 seconds along a highway, but off-road it can only travel half as fast.
Find the area of the region composed of all points the ant can reach in at most 30 seconds.
Note: an ant highway is just a line with no thickness.
Consider the highway system:
+….+….a….+….+
+….b….+….h….+
c….+….A….+….g
+….d….+….f….+
+….+….e….+….+
Key: “A”=Ant; “+”= other crossroads; a, b,… h =
reachable highway-bound crossroad destinations.
Note: if Ant stayed strictly off-road diagonally, she could not reach “b”, “d”, "f" or "h", since the diagonal is of length 1.414m, far greater than 1m
reached off-road in 30 sec..
So the question becomes: if we consider square aceg as the possible outer area, where within can Ant
reach?
I am thinking now of Ant’s range being the superposition of circles formed by
sending Ant up a highway and then going off-road and any angle. The resulting loci of destinations will be something like
a 90 deg-tipped square with each side inwardly-bowed, like a superellipse.
TBC…
Edited on August 17, 2018, 5:55 pm
thoughts
