A man is standing on a rock in the middle of a circular lake of radius 1. There is a tiger on the shore of the lake that can run four times as fast you can swim, however the tiger can not swim. The tiger is hungry and always attempts to keep the distance between the two of you at a minimum.
How can you safely swim to shore?
(In reply to re: Specifically...
I agree with your analysis. Trying to minimize the distance out (4-pi)/4 will get you out the fastest, but with the narrowest margin against the tiger. Approaching 1/4 gets you out the slowest, but gives a better lead against the tiger.
The best path probably is 1: swim out (4-pi)/4, 2: swim in a spiral which reaches 1/4 radius exactly when the tiger is directly opposite of you, 3: head straight for shore.