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### Volume by Inch of a Cone-Shaped Tank

```Date: 03/03/2003 at 21:33:18
From: Matt
Subject: Volume by inch of a cone-shaped tank

I have a chemical tank that has a cone bottom. I can get the total
volume of the cone part. I used your formula under "Frustum of a
Right Circular Cone."

http://mathforum.org/dr.math/faq/formulas/faq.cone.html#conefrustum

I need to know the volume of each inch of the cone. The top radius is
16.25", the bottom radius is 3.25, and the total cone height is 22.5".
I can get the total volume but I need to know the volume as the level
changes. I cannot calculate the top radius as the level changes.
```

```
Date: 03/04/2003 at 01:50:04
From: Doctor Jeremiah
Subject: Re: Volume by inch of a cone-shaped tank

Hi Matt,

What we need is an equation to calculate the top radius at any point.
What we have is a straight line graph like this:

height
|
|                    /
22.5 - - - - - - - - - - + (16.25,22.5)
|                  /
|                 /
|                /
|               /
|              /
|             /
|            /
|           /
|          /
|        / (3.25,0)

Because it is a straight line graph we can use the equation for a
straight line: y=mx+b

For us the "y" is the height and the "x" is the radius so the equation
is really   height = m x radius + b

And we have these two points on the graph:

-----------------------------------------
0       |      3.25
22.5      |     16.25

If we put the first row into height = m x radius + b we get:

height = m x radius + b
0   = 3.25 m + b

If we put the second row into height = m x radius + b we get:

height = m x radius + b
22.5  = 16.25 m + b

If we solve these two equations we get:

m = 22.5/(16.25-3.25)
b = -3.25 x 22.5/(16.25-3.25)

Which gives us this equation for a straight line:

height = 22.5/(16.25-3.25) x radius - 3.25 x 22.5/(16.25-3.25)

But it's all backward. We need an equation with radius = ...

radius = 3.25 + height x (16.25-3.25)/22.5

Now, this kind of makes sense because with a height of 0 we get radius
= 3.25 and with a height of 22.5 we get radius = 3.25 + (16.25-3.25)

So that all means that the radius of the liquid at any height is:

radius = 3.25 + height x (16.25-3.25)/22.5

Which you can use as the top radius of your frustum of a cone.

- Doctor Jeremiah, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Higher-Dimensional Geometry
High School Linear Equations

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