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.
Saying x^3 = 1 - x^3 implies that half of the water is lost and that all of the remainder goes into the cube. This cannot be the case. Some of the water will remain between the larger bucket and the cube that fits inside it.
I think you derived your equation 2s^3 + s = 1 from 1-s^3 = s^3+s
but s^3 is the volume of the cube and s is the total volume of water (inside and out) to the same depth. You counted some water twice.
(1-x^2)(1-x) implies the thickness of the water that will flow into the cube is only the width of the cube. In actuality _all_ water above the cube will flow into it.
Posted by Jer
on 2010-02-03 16:31:24