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Drop into the Bucket (Posted on 2010-02-03) Difficulty: 3 of 5
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.

  Submitted by Jer    
Rating: 4.0000 (3 votes)
Solution: (Hide)
Clearly a volume of x^3 will be lost to spillage where x is the side of the cube. This leaves 1-x^3 of which only a portion will enter the cube.

Without assuming the cube is filled there will still be some volume outside of the cube but in the bucket = x - x^3

This leaves [1-x^3]-[x-x^3] = 1-x for the volume inside the cube. It seems at first that we make x be small but the cubes volume is also given by x^3 so we also need x to be big.

The solution is to set them equal (it turns out we do fill the cube) 1 - x = x^3
x^3 + x - 1 = 0
The real root is about x = .6823278038

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(2): No SubjectDej Mar2010-02-08 19:41:26
re: No SubjectCharlie2010-02-08 12:09:24
SolutionNo SubjectDej Mar2010-02-08 06:11:23
re: Solution - Equation 2?Kenny M2010-02-03 20:26:15
Solutionspoiler now, hopefullyCharlie2010-02-03 16:53:49
Hints/TipsAll 3 below are incorrect. No spoilers.Jer2010-02-03 16:31:24
SolutionSolutionBractals2010-02-03 14:40:17
SolutionsolutionCharlie2010-02-03 14:21:56
solutionDaniel2010-02-03 12:02:16
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