s67 - full. After the first part of the first step, the...

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Solution to Problem 67 Congratulations to this week’s winner Scott Peters Correct solutions were sent in from Bradley students Kenny Koch, Michelle McCluer, Nathan Pauli, Ray Kremer, Kyle Ambroso, Paul Leisher; a partial solution was submitted by Courtney Pelowski. Correct solutions were also received from Cyril Terakopiantz, Tim Kelley (two solutions), Philippe Fondanaiche, William Troxel, Robert McQuaid, Aaron Kahn, Burkart Venzke, Anuradha Ratnaweera, William Webb, Tushar Sharma. A extraspecial mention to Gregory Falcon for his most excellent solution! The answer is 1/e. The following presentation is due to Tim Kelley. First, imagine a scenario similar to but not quite the same as the one described in the problem, where for some positive integer k you do the following: Let 1/k cups leak from the bottom cup. After this has happened, Let 1/k cups leak from the top cup. Repeat these steps k times. At the end of the experiment, the top cup will be empty and the bottom cup will be
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Unformatted text preview: full. After the first part of the first step, the bottom cup will have 1 - (1/k) = (k-1)/k cups of water remaining. The addition of the wine from the top cup in the secod part of the first step will not change this amount. Each succesive cycle of the two steps will result in the amount of water remaining being diminished by 1/k of what was there, so the amount remaining is, again, 1 - (1/k) of the previous amount. This means that after n steps, you have [ (k-1)/k ] n cups of water remaining in the bottom cup. Specifically, after all k steps are done, youll have [IMAGE] cups remaining in the bottom cup. To complete the problem, take k to be very large. The value above will approach the value of the limit, which can be computed exactly using, for example, LHpitals Rule. This value is 1/e....
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This note was uploaded on 03/01/2010 for the course MATH 301 taught by Professor Albertodelgado during the Spring '10 term at Bradley.

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s67 - full. After the first part of the first step, the...

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