On a related note: My grandfather -- who was a pharmacist -- once taught me how to shake a bottle containing two liquids you want to mix together well. Assume the bottle is a cylinder that fits in your hand. You hold it in your hand horizontally, so the bottle fits between your thumb and pinky. Now vigorously rotate your wrist back and forth, so the liquids are forced to crash into each end of the bottle alternately. This works a lot better than holding it vertically. --Dan On 2014-01-28, at 8:48 PM, James Propp wrote:

What's the optimum amount of water to put in a jar if you want to clean the jar by putting some water in it and shaking it for a bit?

I grant that the problem is ill-posed in all kinds of ways, but maybe some not-too-farfetched model gives results in line with experimental data conducted on jars of ordinary shape containing food residues of an ordinary kind.

I seem to recall that number theorist David Solomon, back when we were housemates in Berkeley in the 1980s, came up with an idealization of the problem for which the optimum volume of water was 1/e times the volume of the jar. I'll ask him if he remembers any details.

Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun