The old man and the dog
A man takes his dog for a walk on a sufficiently large path. He decides to throw a stick once so that the dog runs and catches it and brings it back afterwards.
Assume that the trow distance is always the same, that the man and the dog walk/run with a constant speed and that the man walks from left to right.
Where should he throw his stick to ensure that the dog runs the maximal distance?
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1 solution
This is more of a geometrical solution then a calculus one:
Assuming that the distance the stick travels is the same regardless of which direction it is thrown, we can simply compare the distances the dog has to travel to get back to the man.
While it is fairly obvious from the picture which distance is the shortest, we know that if the man is walking from left to right, he should throw the stick in the direction which will cause the stick to land as far to the left as possible for the dog to run the farthest. This will clearly happen when he throws it b a c k w a r d .