Volume of a Cylinder/Tank

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In summary, the conversation discusses the development of a function for the volume of a half-cylinder (the top left piece) as a function of its height, using the radius and height of the cylinder as variables. Various equations and methods are mentioned, such as the area of a segment of a circle and numerical integration, but the desired result is a simpler and more accurate equation for finding the volume. The conversation concludes with a suggestion to look up the ArcCos[Z] part and use integration by parts for the other term.
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Homework Statement


You have a cylinder standing upright. Now, imagine a plane cutting the cylinder in half diagonaly from bottom left to top right.

Develop a function for the volume of the half-cylinder (the top left piece) as a function of its height. Volume would be liquid rising from the bottom of the piece up to the top.

The only variables I have are radius of the cylinder, and height of the cylinder.

This isn't a homework problem... it's an oddly shaped tank I need to develop a curve for, but I figured this is the best place to post.


Homework Equations



V = Pi*R^2*H -not very usefull

A = 1/2 * R^2 * (Theta - Sin(theta)) -This is the area of the segment of a circle made by a chord, where theta is the angle formed by drawing two radii from the center of the circle to each end of the chord.

The Attempt at a Solution



For a solution I'd really like a simple equation and/or something I can reference from a good source.

The way I did it is the following (I'd like something more simple though if it's possible):

Imagine the top view of the half-cylinder tank. As water rises from the bottom, you will see a cross sectional area fill that is equal to an area segment created by a chord. You can take that area and multiply it by a differential height then integrate over the height of the tank.

So I take the equation for the area segment: A = (1/2) * R^2 * (Theta - Sin(Theta))

Theta would start at 0 and increase to 2Pi as the tank fills. However, I need to turn this equation into a function of height (y):

Theta = 2*ArcCos[1-(2y/H)] where H = total height of the tank and y = height

Plug in the new value for theta, and simplify:

A = (1/2) * R^2 * ((2*ArcCos[1-(2y/H)]) - ((1-(2y/H))*(1-(1-(2y/H)^2)^.5))

To make this easier to read I'll consider Z = 1-(2y/H) and substitute it in (I can't use latex on this browser, sorry):

A = (1/2) * R^2 * ((2*ArcCos[Z]) - (Z*(1-Z^2)^.5))

This equation can then be integrated from 0 up to y (where y would be the height of interest).

Solving that integral analytically is very hard (impossible?). I used Excel to numerically integrate, but it is kind of sketchy. I'd like a more simple equation that gives an exact answer if it exists. Thanks for any help.
 
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  • #2
equality of the circle:
[tex]r^{2} = x^{2} + y^{2}[/tex]
solving it for y, choosing the positive one gives the function for the positive half:
[tex] f(x) = \sqrt{r^{2} - x^{2}}[/tex]
integrating it from -r, and doubling is should give us the area of a cross section. Now you should only find the function between h and x, and integrate it again to get the volume.
 
  • #3
BishopUser said:
A = (1/2) * R^2 * ((2*ArcCos[Z]) - (Z*(1-Z^2)^.5))

This equation can then be integrated from 0 up to y (where y would be the height of interest).

Solving that integral analytically is very hard (impossible?). I used Excel to numerically integrate, but it is kind of sketchy. I'd like a more simple equation that gives an exact answer if it exists. Thanks for any help.

You can just look up the part with ArcCos[Z].
The other term can be done with substituting Z = sin U and then integration by parts
 

1. How do you find the volume of a cylinder/tank?

To find the volume of a cylinder or tank, you need to use the formula V = πr²h, where V is the volume, π is the mathematical constant pi, r is the radius of the base of the cylinder, and h is the height of the cylinder. Plug in the values for r and h and use a calculator to solve for V. The resulting unit will be in cubic units, such as cm³ or m³.

2. What are the units for volume?

Volume is measured in cubic units, such as cubic centimeters (cm³) or cubic meters (m³). These units represent the amount of space that an object or substance occupies.

3. Can you find the volume of a cylinder/tank if you only know the height and diameter?

Yes, you can find the volume of a cylinder or tank if you know the height and diameter. The formula for volume in this case is V = π(d/2)²h, where V is the volume, π is pi, d is the diameter, and h is the height. Simply plug in the values for d and h and solve for V.

4. How is the volume of a cylinder/tank related to its capacity?

The volume of a cylinder or tank is directly related to its capacity. Capacity refers to the maximum amount of space that a container can hold. The volume of a cylinder or tank is the measurement of the space inside the container, so it is essentially the same as the capacity.

5. Can the volume of a cylinder/tank change?

Yes, the volume of a cylinder or tank can change if its dimensions or shape change. The volume is directly proportional to the height and radius/diameter of the cylinder or tank, so if any of these values change, the volume will also change. However, the volume will not change if only the substance inside the container changes.

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