SOLUTION: how do i calculate the internal volume of a tapered bucket?

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Question 356351: how do i calculate the internal volume of a tapered bucket?
Found 2 solutions by Alan3354, Fombitz:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
By tapered, do you mean conical?
If so, the volume of a cone is
V+=+pi%2Ar%5E2%2Ah%2F3
A bucket is a frustum of a cone, that is, it doesn't extend to a point.
R1 = radius of top
R2 = radius of bottom
h = depth
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V+=+h%2Api%2A%28R1%5E2+%2B+R2%5E2+%2B+R1%2AR2%29%2F3

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let's look at a cone (in cross section).

Actually there are two cones in the picture.
The large cone with a radius of r2 and height of h1%2Bh2 and a smaller cone with a radius of r1 and height h1. The figure in between would be the tapered bucket if you rotated the cross section about the axis.
So the volume of the tapered bucket would be the volume of the large cone minus the volume of the small cone.
The volume of a cone is,
V%5Bbc%5D=%28pi%2F3%29R%5E2%2AH
Big Cone:
V%5Bsc%5D=%28pi%2F3%29r2%5E2%2A%28h1%2Bh2%29
Small Cone:
V=%28pi%2F3%29r1%5E2%2A%28h1%29
Tapered Bucket:
V%5Btp%5D=%28pi%2F3%29r2%5E2%2A%28h1%2Bh2%29-%28pi%2F3%29r1%5E2%2A%28h1%29
There is also a relationship between the radii and the heights.
%28r1%29%2F%28h1%29=%28r2%29%2F%28h1%2Bh2%29
r1%2Ah1%2Br1%2Ah2=r2%2Ah1
h1%28r2-r1%29=r1%2Ah2
h1=+%28r1%2Ah2%29%2F%28r2-r1%29+
h1%2Bh2=%28r1%2F%28r2-r1%29%29%2Bh2
h1%2Bh2=%28r1%2F%28r2-r1%29%29h2%2Bh2%2A%28%28r2-r1%29%2F%28r2-r1%29%29
h1%2Bh2=+%28r2%2F%28r2-r1%29%29%2Ah2
Substituting,
V%5Btp%5D=%28pi%2F3%29%28r2%5E2%2A%28h1%2Bh2%29-r1%5E2%2A%28h1%29%29

V%5Btp%5D=%28pi%2F3%29%28h2%2F%28r2-r1%29%29%28r2%5E3-r1%5E3%29%29
Dividing by %28r2-r1%29
highlight%28V%5Btp%5D=%28pi%2F3%29%28h2%29%28r2%5E2%2Br1%2Ar2%2Br1%5E2%29%29%29