An ant starts at the origin of the coordinate plane. The ant only has enough energy to walk for 10 units along either the x- or y-axis before stopping. These lines are the two ant highways.

However, if it goes off the highway, more energy is needed for the same distance. That is, the further the ant deviates, the greater the energy consumption, as represented by the factor 1+0.4d, where d is the distance to the nearest highway. For example, if the ant is 1 unit away from the nearest highway, it uses 1.4 times the energy on the highway to walk 1 meter.

What is the area of the set of points the ant can reach before getting tired and stopping?

(In reply to possible analytical approach (thoughts) by Steven Lord)

I don't think a straight line will lead to a maximal distance. If you take the endpoints of the curves shown in my second illustration, I don't think you'll find any straight-line path off the road (that is, follow the road for a while, then go off on a straight line at some angle) that will allow the ant to arrive there.

Posted by Charlie on 2018-08-26 12:49:47