Volume by Inch of a Cone-Shaped TankDate: 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) | / | / | / | / | / | / | / | / | / ----------------------+---------+---------------> radius | / (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: height | radius ----------------------------------------- 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/ |
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