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 re(2): heuristic computer solution by Charlie)

The avoidance of counting overlapping rectangles in the numeric integration seems daunting enough, as the endpoints are determined not by going directly into the x-value, but after an integration of positions, but an additional complication is that extra area may be added by starting near (how near I don't know) the origin to go off at angles other than zero (that is, other than tangent to the axis). And do we need to go from the perpendicular road as well, but wind up in the octant in which we're interested. It sounds as if we need to merge several sets of I-dont-now-how-many x-values into a set in which the highest y-value is obtained.

Posted by Charlie on 2018-08-26 12:44:59