Volume of a barrel.

[ggauny@live.fr]

Hi,
In learning in hight level mathematic, we have an exercise it is how calculate the
capacity of a barrel ?
In internet is billion solution, but never the same volume is found for the same barrel !!!! ????
Is people have the good formula that is realy exact ?
It is very hard because they speak "parabola, eliptical generatrice" or cos, sin etc.

[jch]

Bonjour Gérard,

Voilà un bon problème que tu nous soumets.
A mon avis, le point crucial est la modélisation du profil de cintrage de la douve (génératrice ?), ellipse, parabole, et je cherche encore pour la formule de 1865...

On reste dans le cas général du calcul du volume d'un cylindre (Pi*r²*h) et toute la difficulté est dans l'estimation du rayon moyen du tonneau.

L'approche est bien résumée dans les deux documents que tu as trouvé et passe par l'intégrale (sommation) du profil sur 1/2 hauteur (on a une symétrie sur cet axe).

Accroche toi, je regrette parfois de n'avoir pas approfondi certaines notions quand j’étais à l'I.U.T, mais je ne désespère pas de m'y remettre, quand j'aurais plus de temps pour cela. En attendant, j'essaye de suivre à minima le programme scolaire de ma fille...
Bon courage.
Jean-Christophe.

Hi Gérard,

Thanks for this challenge.
IMHO, the main point is about estimation of the profile of the barrel.
I may be solved using either the classical cylinder formula (Pi*r²*h) with an estimate of a medium radius or by integrating the profile (generatrice ?) along half the length of the barrel.

Hope this may help you.
Best regards.
Jean-Christophe.

[Dieter]

(03-20-2016 08:25 AM)ggauny@live.fr Wrote:  In learning in hight level mathematic, we have an exercise it is how calculate the
capacity of a barrel ?
In internet is billion solution, but never the same volume is found for the same barrel !!!! ????

Sure. The exact volume depends on the exact shape of the barrel.

(03-20-2016 08:25 AM)ggauny@live.fr Wrote:  Is people have the good formula that is realy exact ?
It is very hard because they speak "parabola, eliptical generatrice" or cos, sin etc.

The is no general formula because the volume depends on how the shape of the barrel is defined, i.e. the question of how wide the barrel is at a given height. The paper you attached assumes that the arc that defines the outline of the barrel is a parabola. In your paper this is the relation between x and y. If this can be described as a parabola (y=ax²+c), the final formula is fine.

For a general solution you can only say that the volume equals the integral that is also mentioned in your paper. This leads to Guldin's second rule: the volume is Pi times the integral of the square of the function y(x).

Long time ago all this lead to what is today known as Simpson's rule for numerical integration: Johannes Kepler defined what in German is called "Keplersche Fassregel" ("Kepler's barrel rule"). He had exactly the same problem with the (more or less) exact volume of wine barrels. ;-)

Finally, the method given in your attached PDF is another approximation method. For the given example the three methods (exact, assuming a parabola shaped barrel, Kepler's rule and the one in the PDF) return a volume of 0,7958 resp. 0,7927 and 0,7907 cubic meters.

Dieter

[Tugdual]

(03-20-2016 08:25 AM)ggauny@live.fr Wrote:  Hi,
In learning in hight level mathematic, we have an exercise it is how calculate the
capacity of a barrel ?
In internet is billion solution, but never the same volume is found for the same barrel !!!! ????
Is people have the good formula that is realy exact ?
It is very hard because they speak "parabola, eliptical generatrice" or cos, sin etc.

So what is your question eventually?

[ggauny@live.fr]

Hi,
My question is how exactely find the exact capacity of a barrel and why it is not a unique answer, and why so many equations ?
Me I dont' know what is parabola or eliptic curve, so for my excercise to do for
I encounter problems of course.
Dieter was speaking of Kepler, so I had remember in my "very" old book of geometrie, when I was in school before go to work there is a solution without complicated formulas.
I will do my excercise like in this time.

[ggauny@live.fr]

It is said that this formula is STRICTELY EXACT.

[ggauny@live.fr]

In my prime younght, in my grand'mother little farm in Ancemont village of Meuse, when she bought a barrel she ask me to verify the capacity. In this time I simply fill
the barrel with a water-pump and his counter in litres, décilitres and centilitres. When barrel is full, I see the counter and I say to grand'mother. She was very
suspicious wary and this method say if the dealer is a liar on capacity of the barrel (to have more money) and second this method say if the barrel is water-proof : very important you know.
It is not "mathematic" but very effective !!!!! only with not bigs barrels of course.

I will not write this on my copy for the professeur....

[Tugdual]

(03-21-2016 07:44 AM)ggauny@live.fr Wrote:  Hi,
My question is how exactely find the exact capacity of a barrel and why it is not a unique answer, and why so many equations ?
Me I dont' know what is parabola or eliptic curve, so for my excercise to do for
I encounter problems of course.
Dieter was speaking of Kepler, so I had remember in my "very" old book of geometrie, when I was in school before go to work there is a solution without complicated formulas.
I will do my excercise like in this time.

A cylinder is a barrel with flat sides but immediately you suspect it cannot have the same volume as a wine barrel. And here is the answer to your question: there is no such thing as a unique barrel, hence there are multiple answers.
https://fr.m.wikipedia.org/wiki/Tonneau_(formules)
The generic way to calculate the volume of a barrel is to cut it into a stack for extremely thin slices, so thin that you might consider that each slice is a cylinder. You then sum the volume of each cylinder to achieve the entire barrel volume. A thin cylinder height is called dx (where x is travelling on the vertical axis of the cylinder) and the summing operation is called an integration. Solving the problem is as simple as resolving an integral equation.

Remember that when attending math class, the current lesson is giving you clues about how to solve the problem. Are you learning integral equations?

[Dieter]

(03-21-2016 07:46 AM)ggauny@live.fr Wrote:  It is said that this formula is STRICTELY EXACT.

The formula in section 552 of the old book indeed is exact... IF (and only if) the shape of the barrel is an ellipse. ;-) This formula translates to Pi*h/12*(2D²+d²) if d and D are the two diameters (at the end and in the middle) of the barrel. This is exactly the formula that is mentioned on the website posted by Tugdual – for barrels that are shaped like an ellipse. ;-)

NO, this formula is NOT exact for ANY barrel. That's not possible. All this is very easy to understand: take a look at the graphics of the barrel in any of these papers. If you only have the end diameters and the one in the middle, you can construct an infinite number of different barrels that are, say, 50 cm at the end and 60 cm in the center. On one of them the shape is similar to an ellipse, on others it's more like a parabola, and you can also draw a barrel that has another shape that cannot be defined by any standard geometric figure. And of course they all have different volumes, even if the diameters at the end and in the middle are the same. They are only the same at these three points.

Gérard, please take a look at the wikipedia website Tugdual posted. This site says it all. Or try that one. Take a look at the diagram. The lines AFB resp. DEC can slightly differ even if the diameters d and D are the same. So the volumes are slightly different as well. Which is exactly the point here.

Yes, there is a general and exact solution. That's the one with the integral. Here you need the function that defines the line AFB or DEC. Please read the chapter "Calcul".

Dieter

[jch]

(03-21-2016 04:40 AM)BarryMead Wrote:  

(03-20-2016 03:11 PM)jch Wrote:  I may be solved using either the classical cylinder formula (Pi*r*h).

I am sure you meant to say the volume of a cylinder is (Pi * r² * h).

Take Care, Barry

Of course !
Many thanks for the correction Barry.
Original post edited.

[Tugdual]

By the way, I think this thread has nothing to see with hp calculators.
Could be good to move it in "not hp caclculators"

[Dieter]

(03-22-2016 06:38 AM)Tugdual Wrote:  By the way, I think this thread has nothing to see with hp calculators.
Could be good to move it in "not hp caclculators"

It should be moved to the General Forum. That's where it belongs:
"...Also HP news, general math and science etc."

Dieter

[ggauny@live.fr]

Hi,
Thanks for explanations and advertising. How can I move my thread to other
place ?

[Dieter]

(03-22-2016 07:56 AM)ggauny@live.fr Wrote:  Thanks for explanations and advertising. How can I move my thread to other
place ?

You can't. This can only be done by the administrators, usually after a request in the "Forum Issues and Administration" forum. But as you can see, this thread already has been moved to the General Forum.

Dieter

[Tugdual]

(03-22-2016 07:39 PM)Dieter Wrote:  

(03-22-2016 07:56 AM)ggauny@live.fr Wrote:  Thanks for explanations and advertising. How can I move my thread to other
place ?

You can't. This can only be done by the administrators, usually after a request in the "Forum Issues and Administration" forum. But as you can see, this thread already has been moved to the General Forum.

Dieter

Greetings to the moderators for the amazing work...