A river flows from town A to town B and has the property that any point on either of its banks is no...
geometry/weird.river New
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Problem
A river flows from town A to town B and has the property that any point on either of its banks is no farther than 100 yards from some point on the other bank. [one way to see what this property means: if you had a 100 yard long portable bridge, you would be able to stand *anywhere* along _either_ bank and always be able to set up your bridge to cross the river]. a) A boat sails down the river, trying to stay always within a distance _d_ from both banks. For what values of _d_ can we guarantee that such a trip is possible. b) How far can a swimmer in the river be from the nearest bank? -- bernie@fantasyfarm.com
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>a) A boat sails down the river, trying to stay always within a distance >_d_ from both banks. For what values of _d_ can we guarantee that such >a trip is possible. Clearly we can follow one bank if d=100. I will construct a river which requires d=100. >b) How far can a swimmer in the river be from the nearest bank? Arbitrarily far. The required river is a little hard to draw in ASCII. I will describe the land rather than the river. To construct the right bank, start with a flat coastline, attach a thin jetty. The jetty branches like a T. The two arms curve around a large radius R to almost meet at the other end where the opening is ~200 yards. To construct the left bank, start with a flat coastline. Dig a channel inland to connect to a circular lake of radius ~R+100. Now put the two banks together just right, and adjust a few distances. A swimmer in the center is far from any bank, and for the boat to pass the opening, it must have d=100 regardless of whether it goes straight across, or follows the inside of the big circle. -- Dave Ring <dwr2560@tam2000.tamu.edu> (May 1995)