October 21, 2004 was Martin Gardner's 90

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.