Consider a bucket in the shape of a cube 1 foot on a side and filled with water.

A smaller cube shaped container, open at the top, is pushed straight down into the bucket without rotating it. At first it displaces some water which spills out of the bucket but when this container is pushed down far enough the extra water will pour into it.

If this container is very small it will be completely filled and sink to the bottom. If it is very big it will not end up with much water in it. What dimensions of this cubic container will maximize the volume that ends up inside of it.

(In reply to Solution by Bractals)

While your method is correct, you equation (2) doesn't match your intent.  The surface area of the water is 1 sqaure unit, times the distance the smaller cube can be pushed down, X², then leads to the same answer that Charlie found in Comment #5, which is correct.

Posted by Kenny M on 2010-02-03 20:26:15