Mixing
Using Area of Segment - Derive General Formula for the Volume of a Tilted Cylinder Partially Filled with...
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I would be very grateful if I could get some advice as to where I am going wrong! I am trying to derive the formula to calculate the volume of a partially filled cylinder and it just doesn't seem to be working.
I have attached my work here in the onedrive link (files are numbered according to pages). Eventually I want to do this for a cylinder inside a cylinder, so you'll see I have subscripts "h" for the outer cylinder. (The final goal will be to subtract the "l"/inner cylinder volume from the "h" volume, however, I am not at that point yet, and the only place you will see the subscript "l" is in the set up drawing). My goal right now is to just see if I can get the outer cylinder volume (the "h" cylinder) and I will worry about subtracting the inner cylinder volume later.
Seeing as I am just focusing on the outer/"h" cylinder, I ended up dropping all of my subscripts as you'll see in my work. It was just getting too tedious.
Here are the variables:
R(h) - radius of cylinder (tunnel) - CONSTANT
h(g) - height of water, measured from the lowest part of the cylinder - CONSTANT
y(h) - elevation of the bottom of the cylinder (tunnel) - Derived to be y = sx - sl
s(l) - slope of cylinder - CONSTANT
l(t) - length of cylinder (tunnel) - CONSTANT
x - distance along x-axis (horizontal distance from the highest part of the cylinder)
***Everything in brackets for the variable list is a subscript that I eventually drop.
***Note that my work is only valid assuming the water level reaches at least to the bottom of the upper face.
Thanks so much in advance, if you are able to detect where I am going wrong!
Here is my work:
https://1drv.ms/f/s!AvU6fPuuMW9OplShL2tfqNlrhno7
calculus integration definite-integrals volume trigonometric-integrals
asked Jan 26 at 9:27
Royal RougeRoyal Rouge
13
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add a comment |
$begingroup$
I would be very grateful if I could get some advice as to where I am going wrong! I am trying to derive the formula to calculate the volume of a partially filled cylinder and it just doesn't seem to be working.
I have attached my work here in the onedrive link (files are numbered according to pages). Eventually I want to do this for a cylinder inside a cylinder, so you'll see I have subscripts "h" for the outer cylinder. (The final goal will be to subtract the "l"/inner cylinder volume from the "h" volume, however, I am not at that point yet, and the only place you will see the subscript "l" is in the set up drawing). My goal right now is to just see if I can get the outer cylinder volume (the "h" cylinder) and I will worry about subtracting the inner cylinder volume later.
Seeing as I am just focusing on the outer/"h" cylinder, I ended up dropping all of my subscripts as you'll see in my work. It was just getting too tedious.
Here are the variables:
R(h) - radius of cylinder (tunnel) - CONSTANT
h(g) - height of water, measured from the lowest part of the cylinder - CONSTANT
y(h) - elevation of the bottom of the cylinder (tunnel) - Derived to be y = sx - sl
s(l) - slope of cylinder - CONSTANT
l(t) - length of cylinder (tunnel) - CONSTANT
x - distance along x-axis (horizontal distance from the highest part of the cylinder)
***Everything in brackets for the variable list is a subscript that I eventually drop.
***Note that my work is only valid assuming the water level reaches at least to the bottom of the upper face.
Thanks so much in advance, if you are able to detect where I am going wrong!
Here is my work:
https://1drv.ms/f/s!AvU6fPuuMW9OplShL2tfqNlrhno7
calculus integration definite-integrals volume trigonometric-integrals
asked Jan 26 at 9:27
Royal RougeRoyal Rouge
13
$endgroup$
$begingroup$
I've continued working on this... I have re-written, hopefully more clearly, the first part, in my attempt to re-check yet again. This may be a little less daunting to look at. The first part is just the integral used to calculate the volume of the section (from which the volume of the triangle would have to later be subtracted). But even the volume of the section is coming back as unreasonable when I input some example numbers. See: 1drv.ms/f/s!AvU6fPuuMW9OpmCbCNz8j_jcFO9j
$endgroup$
– Royal Rouge
Jan 27 at 19:09
$begingroup$
Note, I have completely removed subscripts here, right from the start.
$endgroup$
– Royal Rouge
Jan 27 at 19:10
add a comment |
$begingroup$
I would be very grateful if I could get some advice as to where I am going wrong! I am trying to derive the formula to calculate the volume of a partially filled cylinder and it just doesn't seem to be working.
I have attached my work here in the onedrive link (files are numbered according to pages). Eventually I want to do this for a cylinder inside a cylinder, so you'll see I have subscripts "h" for the outer cylinder. (The final goal will be to subtract the "l"/inner cylinder volume from the "h" volume, however, I am not at that point yet, and the only place you will see the subscript "l" is in the set up drawing). My goal right now is to just see if I can get the outer cylinder volume (the "h" cylinder) and I will worry about subtracting the inner cylinder volume later.
Seeing as I am just focusing on the outer/"h" cylinder, I ended up dropping all of my subscripts as you'll see in my work. It was just getting too tedious.
Here are the variables:
R(h) - radius of cylinder (tunnel) - CONSTANT
h(g) - height of water, measured from the lowest part of the cylinder - CONSTANT
y(h) - elevation of the bottom of the cylinder (tunnel) - Derived to be y = sx - sl
s(l) - slope of cylinder - CONSTANT
l(t) - length of cylinder (tunnel) - CONSTANT
x - distance along x-axis (horizontal distance from the highest part of the cylinder)
***Everything in brackets for the variable list is a subscript that I eventually drop.
***Note that my work is only valid assuming the water level reaches at least to the bottom of the upper face.
Thanks so much in advance, if you are able to detect where I am going wrong!
Here is my work:
https://1drv.ms/f/s!AvU6fPuuMW9OplShL2tfqNlrhno7
calculus integration definite-integrals volume trigonometric-integrals
asked Jan 26 at 9:27
Royal RougeRoyal Rouge
13
$endgroup$
I would be very grateful if I could get some advice as to where I am going wrong! I am trying to derive the formula to calculate the volume of a partially filled cylinder and it just doesn't seem to be working.
I have attached my work here in the onedrive link (files are numbered according to pages). Eventually I want to do this for a cylinder inside a cylinder, so you'll see I have subscripts "h" for the outer cylinder. (The final goal will be to subtract the "l"/inner cylinder volume from the "h" volume, however, I am not at that point yet, and the only place you will see the subscript "l" is in the set up drawing). My goal right now is to just see if I can get the outer cylinder volume (the "h" cylinder) and I will worry about subtracting the inner cylinder volume later.
Seeing as I am just focusing on the outer/"h" cylinder, I ended up dropping all of my subscripts as you'll see in my work. It was just getting too tedious.
Here are the variables:
R(h) - radius of cylinder (tunnel) - CONSTANT
h(g) - height of water, measured from the lowest part of the cylinder - CONSTANT
y(h) - elevation of the bottom of the cylinder (tunnel) - Derived to be y = sx - sl
s(l) - slope of cylinder - CONSTANT
l(t) - length of cylinder (tunnel) - CONSTANT
x - distance along x-axis (horizontal distance from the highest part of the cylinder)
***Everything in brackets for the variable list is a subscript that I eventually drop.
***Note that my work is only valid assuming the water level reaches at least to the bottom of the upper face.
Thanks so much in advance, if you are able to detect where I am going wrong!
Here is my work:
https://1drv.ms/f/s!AvU6fPuuMW9OplShL2tfqNlrhno7
calculus integration definite-integrals volume trigonometric-integrals
calculus integration definite-integrals volume trigonometric-integrals
asked Jan 26 at 9:27
Royal RougeRoyal Rouge
13
asked Jan 26 at 9:27
Royal RougeRoyal Rouge
13
edited Jan 27 at 19:13
Royal Rouge
asked Jan 26 at 9:27
Royal RougeRoyal Rouge
13
asked Jan 26 at 9:27
Royal RougeRoyal Rouge
13
asked Jan 26 at 9:27
Royal RougeRoyal Rouge
13
13
$begingroup$
I've continued working on this... I have re-written, hopefully more clearly, the first part, in my attempt to re-check yet again. This may be a little less daunting to look at. The first part is just the integral used to calculate the volume of the section (from which the volume of the triangle would have to later be subtracted). But even the volume of the section is coming back as unreasonable when I input some example numbers. See: 1drv.ms/f/s!AvU6fPuuMW9OpmCbCNz8j_jcFO9j
$endgroup$
– Royal Rouge
Jan 27 at 19:09
$begingroup$
Note, I have completely removed subscripts here, right from the start.
$endgroup$
– Royal Rouge
Jan 27 at 19:10
add a comment |
$begingroup$
I've continued working on this... I have re-written, hopefully more clearly, the first part, in my attempt to re-check yet again. This may be a little less daunting to look at. The first part is just the integral used to calculate the volume of the section (from which the volume of the triangle would have to later be subtracted). But even the volume of the section is coming back as unreasonable when I input some example numbers. See: 1drv.ms/f/s!AvU6fPuuMW9OpmCbCNz8j_jcFO9j
$endgroup$
– Royal Rouge
Jan 27 at 19:09
$begingroup$
Note, I have completely removed subscripts here, right from the start.
$endgroup$
– Royal Rouge
Jan 27 at 19:10
$begingroup$
I've continued working on this... I have re-written, hopefully more clearly, the first part, in my attempt to re-check yet again. This may be a little less daunting to look at. The first part is just the integral used to calculate the volume of the section (from which the volume of the triangle would have to later be subtracted). But even the volume of the section is coming back as unreasonable when I input some example numbers. See: 1drv.ms/f/s!AvU6fPuuMW9OpmCbCNz8j_jcFO9j
$endgroup$
– Royal Rouge
Jan 27 at 19:09
$begingroup$
I've continued working on this... I have re-written, hopefully more clearly, the first part, in my attempt to re-check yet again. This may be a little less daunting to look at. The first part is just the integral used to calculate the volume of the section (from which the volume of the triangle would have to later be subtracted). But even the volume of the section is coming back as unreasonable when I input some example numbers. See: 1drv.ms/f/s!AvU6fPuuMW9OpmCbCNz8j_jcFO9j
$endgroup$
– Royal Rouge
Jan 27 at 19:09
$begingroup$
Note, I have completely removed subscripts here, right from the start.
$endgroup$
– Royal Rouge
Jan 27 at 19:10
$begingroup$
Note, I have completely removed subscripts here, right from the start.
$endgroup$
– Royal Rouge
Jan 27 at 19:10
add a comment |
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I've continued working on this... I have re-written, hopefully more clearly, the first part, in my attempt to re-check yet again. This may be a little less daunting to look at. The first part is just the integral used to calculate the volume of the section (from which the volume of the triangle would have to later be subtracted). But even the volume of the section is coming back as unreasonable when I input some example numbers. See: 1drv.ms/f/s!AvU6fPuuMW9OpmCbCNz8j_jcFO9j
$endgroup$
– Royal Rouge
Jan 27 at 19:09
$begingroup$
Note, I have completely removed subscripts here, right from the start.
$endgroup$
– Royal Rouge
Jan 27 at 19:10