 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Drop into the Bucket (Posted on 2010-02-03) 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.

 See The Solution Submitted by Jer Rating: 4.0000 (3 votes) Comments: ( Back to comment list | You must be logged in to post comments.) Solution | Comment 3 of 9 | `Let x be the side length of the small cube.`
`The maximum amount of water that the smallcube can hold is `
`       h(x) = x^3                     (1)`
`The maximum amount of water that can spillinto the small cube is the surface area ofthe water times the distance that the cubecan be pushed down`
`       s(x) = (1 - x^2)*(1 - x)       (2)`
`On the interval [0,1], h(x) is strictly increasing and s(x) is strictly decreasing. Therefore, to find the value of x which will maximize the amount of water we seth(x) = s(x) and solve for x,`
`       x^3 = (1 - x^2)*(1 - x)`
`           or`
`       x^2 + x - 1 = 0`
`Thus,`
`       x = (sqrt(5) - 1)/2`
` `

 Posted by Bractals on 2010-02-03 14:40:17 Please log in:

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