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Calculus/Work Problem- Conical Tank

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Expert: - 2/16/2009


Question
Conical Tank is 5 feet wide at the top, 2 feet tall. How much work to fill the entire tank from bottom with syrup. (density 48lbs/ft3).

Answer
The volume of a cone is πr²h/3.
We know that 2r = 5, h = 2 and π=3.1416 (roughly),
so the volume can be found.

Since the volume is in ft³, and density is in lbs/ft³,
multiply this volume by the density and you will get the weight.

That was interesting, but now to answer your question.

Work is distance * weight, so do an integral from 0 to 2 of
the cross sectional weight.  The diameter at the top is 5,
so the radius is 2.5.  This means the ratio of radius to height is
2.5/2 = 1.25.  So at any given depth x, the radius r = 1.25x.
This means that the cross sectional area is πr², which is 1.5625πx².
The weight is then 48*1.5625πx² = 75πx².

Work is distance * weight, so we need to do is find
⌠2
⌡0 x*75πx² dx =

⌠2
⌡0 75πx³ dx = (75π)2^4/4.  2^4=16, 16/4=4, 4*75=300,

so the answer is 300π.

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