## Mathematical Recreations

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This optimization problem can be solved using only algebra and geometry, but it's difficult to do that way. A little bit of calculus is really helpful.

#### Duck Hunt

The following topic is a variation of one of Paul Hsieh's puzzles. Problem 4 of Set 4 in Paul's Mathematical section reads as follows:

A duck is swimming in the center of a perfectly circular pond, and a fox is on the land at the very edge of the pond. The fox is hungry and wants to eat the duck, but it cannot swim. The duck wants to get out of there but needs to reach land in order to be able to fly away. If the duck swims at 1 meter per second, and the fox can run around on land at 4 meters per second, can the duck somehow escape?

The question that I want to consider is only slightly different. How fast does the fox have to be to prevent the duck from escaping?

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Thanks,
Steve