## Jul 14 Rabbit in a Pond

There is a rabbit which has swum into the middle of a circular pond. A hungry wolf prowls around the banks of the pond, wanting to catch the rabbit, but not wanting to get its feet wet. While the rabbit swims slowly (only one quarter of the land speed of the wolf), if it manages to reach the edge of the pond it will be able to get away because it is faster than the wolf on foot.

Can the rabbit escape? Below there is a hint and below it are the answers.

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Hint: what happens if the rabbit just swims straight for the opposite edge from the wolf?

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Answer: if the rabbit swims in a straight line to the opposite side of the pond to the wolf they will have to travel one radius, which we are going to label r.

However the wolf will have to travel halfway around the circumference, which is 0.5*Pi*2r=Pir. So the have Pi times the distance to travel. However, since they are 4 times faster and 4 > Pi, the wolf will make it to the other bank fast enough to catch the rabbit.

However the rabbit can escape. Firstly it swims almost one quarter of the radius out. Then it travels around that circle of radius almost r/4 until it is opposite the wolf (i.e. when you can draw a straight line between the rabbit and the wolf and have it pass through the centre). It is able to do this because it is tracing out a circle with a circumference just less than 4 times smaller than the wolf’s. Once it is opposite it then makes a mad dash for the edge in a straight line. Since it only has 3r/4 left to cover, but the wolf will have Pir like before, we have to compare 3/4 to Pi/4, which makes the rabbit get to the edge just before.