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 All 3 below are incorrect. No spoilers.
When the small cube has just barely reached the point at which water starts to enter it (when the tops of the two cubes are level), s^3 water has spilled out of the larger cube (the bucket). As the smaller cube sinks lower and the water rushes in, replacing the air that had been there the water level goes down by an amount equal to s^3, so when full, indeed, as you say, only s^3 remains above the top level of the smaller cube, so indeed the correct equation would be:
s^3 + s = 1
s = .6
s = (1 - s) ^ (1 / 3)
PRINT s, s ^ 3
s = .6823278038280192 feet as the side length of the small cube.
It's volume is .3176721961719804 cubic feet.
Posted by Charlie
on 2010-02-03 16:53:49