Tuesday, July 8, 2008
Duck in Pond
A duck starts off in the center of a perfectly circular pond with radius r. On the edge of the pond is a cat. The cat can run four times as fast as the duck can swim, but can not enter the water. Is it possible for the duck to reach any edge of the pond without the cat being there to catch it? If no, provide a proof as to why it can not be done. If yes, provide the strategy that the duck must use.
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3 comments:
No, it is impossible. Even if the cat were starting at the opposite end of the point of the duck's destination, the cat would only have to travel 3.14 r (half of the circumference of the pond) at most, while the duck would have to travel at least r. Since the cat is 4 times as fast as the duck, the cat will always catch the duck.
It is true that if the duck is in the center of the pond he can not reach the end before the cat does. However, is there a strategy for the duck to have to cover less distance while the cat still has to travel the entire half length of the circumference?
ANSWER:
Let's say the radius of the pond is 1. The duck should swim towards the cat until it is .24 from the center. Then the duck should start swimming in a perfect circle of radius .24 in one direction over and over. The cat will try to follow, but with each time around the cat will fall slightly behind (because the cat would only be able to follow perfectly around if the radius of the duck's circle was 1/4 the cat's). Once the duck keeps doing this, he will eventually reach a point where the cat is directly behind him through the center of the pond, and the duck will be able to shoot for the closest edge and travel the .76 distance before the cat travels the 3.14 distance around the edge
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