Problem #71

October 21, 2004 was Martin Gardner's 90th birthday. Gardner was the author of the "Mathematical Games" column in Scientific American from 1956 to 1986. Through his columns and books he popularized both recreational and "serious" mathematics and inspired a generations of young people to pursue mathematical careers. A partial list of his mathematical books is here.

In tribute to Martin Gardner, the first part of this month's problem is paraphrased from his book "Wheels, Life, and Other Mathematical Amusements" and the second part is a variant of it.

1. While at a picnic, a man noticed the following. When his can of soda was full, the center of mass of the can and soda was in the middle of the can [note: we are assuming that the can is a perfect cylindrical shell and are ignoring any openings]. After drinking some of the soda and putting the can down, the center of mass is clearly lower, however when he is finished with the soda, the center of mass is back in the center. The first problem is to find the minimum height of the center of mass: if r denotes the ratio of the mass of the soda to the mass of the can, express the minimum height of the center of mass as a fraction of the (internal) height of the can as a function of r..

2. What if the can has a spherical shape with an internal diameter of 6 inches and the ratio of the mass of the soda to the mass of the can is 8:1 (as it was in Gardner's original formulation)? Please give your answer to three decimal places.

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