Minimizing Total Time, Part II

In a famous math problem, explored in "real life" by Tim Pennings, http://www.maa.org/features/elvisdog.pdf, a dog is to run along a straight shore, then jump into the water and start swimming to get to a ball, thrown into the water at some distance down the shore. The goal is to find the point where the dog should leave the shore, in order to minimize the time it takes to get to the ball.

In this second applet, we concentrate on looking at the graph traced out for total time, T, as a function of the jump-off point, J. "Throw" the ball to some fixed place, then trace out the graph of T v. J, for different values of "r" and "s."

(To "clean" the screen, it's easiest to zoom-scroll in and out.)

1. Fix the running speed r, at 6.4 m/sec, and the swimming speed, s, at 0.9 m/sec (as in the paper.) Then trace out the function T(J) for your particular location for the ball. Visually locate the minimum.

2. Find a ratio r/s where locating the optimum seems to be the most valuable. In other words, for large enough values of s compared to r, it would be best just to swim straight to the ball. For large enough values of r compared to s, the optimal point is so close to straight down from the ball as to not matter.

3. Given that the experimenter can't change the dog's running or swimming speeds, what should she/he do, to get clearer evidence of whether the dog is actually following an optimal strategy? (Hint: look at Dr. Pennings' result about the fixed ratio of y to x. How do you make "y" larger, in relation to z?

Dave Matthews, Created with GeoGebra