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.
let the dimensions of the submerging bucket be x.
as you begin to submerge the bucket it displaces a volume of water equal to the volume that has been submerged. This continues up until the point at which the top of both buckets are level with each other. At this point the volume of water remaining is 1-x^3. Thus the amount that will make it into the smaller bucket is equal to the lesser of x^3 and 1-x^3. Thus this amount is maximized when
x^3 = 1-x^3
2x^3 = 1
x^3 = 1/2
x = (1/2)^(1/3)
or approximately 0.793701 ft
my physics is fairly week so I have a sinking feeling that I may have missed something.
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Posted by Daniel
on 2010-02-03 12:02:16 |
solution
