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.

The smaller cube of volume x³ cubic feet, when filled and sunk to the bottom of the bucket after displacing x³ cubic feet of water, such that it is the maximum size to again be filled with x³ cubic feet will have edge length of 1 - x.
Thus, the problem can be stated where x³ = 1 - x, which can be restated as the cubic equation: x³ + x - 1 = 0;  x can then be found by solving the cubic equation.

Using the cubic, quadratic, and linear coefficients and the constant term of the polynomial equation, {1, 0, 1, -1}; the real root for x is solved as 
([1/2 + SQRT(31/108)]^1/3 + [1/2 - SQRT(31/108)]^1/3)
~= 0.6823278038280193273694837397 feet.

 

Edited on February 8, 2010, 8:13 pm

Posted by Dej Mar on 2010-02-08 06:11:23