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Problem of the Week 1138

Surveying on Earth

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Alice and Bob own roughly rectangular pieces of land on Earth, which is assumed to be a perfect sphere of radius 3950 miles.

Alice's land is bounded by four fences, two of which run in an exact north-south direction and two of which run in an exact east-west direction. Her north-south fences are exactly 10 miles long; her east-west fences are exactly 20 miles long.

Bob's land is similarly bounded by four fences, but his north-south fences are 20 miles long and his east-west fences are 10 miles long.

Whose land has the greater area?

This problem is reminiscent of the classic "What color is the bear?" puzzle. It seems impossible to answer at first, but repays careful thought. Just ask yourself: "What color is their bear?"

Source: Due to Lester R. Ford, an avid problemist who was editor of the American Mathematical Monthly during World War II and, later, president of the Mathematical Association of America.

© Copyright 2010 Stan Wagon. Reproduced with permission.

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15 November 2010