How many marbles in a cylinder if

[LearninDaMath]

Homework Statement

Suppose there are 1000 marbles/meter^3

And I have a cylinder of \pir^2(h), where radius is 2 meters and height is 5 meters.

How many marbles can the cylinder hold?

Homework Equations

Do I have to somehow convert volume of the form \pir^2(h) to volume of the form meters^3? If so, how can this be done? For example, if I have a cylinder of dimensions \pi3^2(10), how can I describe this volume in terms of meters^3?

The Attempt at a Solution

This is not an assigned homework question, just self study.

 

[SammyS]

Homework Statement

Suppose there are 1000 marbles/meter^3

And I have a cylinder of \pir^2(h), where radius is 2 meters and height is 5 meters.

How many marbles can the cylinder hold?

Homework Equations

Do I have to somehow convert volume of the form \pir^2(h) to volume of the form meters^3? If so, how can this be done? For example, if I have a cylinder of dimensions \pi3^2(10), how can I describe this volume in terms of meters^3?

The Attempt at a Solution

This is not an assigned homework question, just self study.

(meters) times (meters) times (meters) is meters³ .

 

[LearninDaMath]

(meters) times (meters) times (meters) is meters³ .

Yes, I know what the exponent of 3 means. My question is in regard to a cylinder.For instance, when I compute πr^2(h) when r=3 and h=10, I get 282.74... but what does that actually mean? Is it 282.74 m^3? Or do I need to do some further converting?

 

[Ray Vickson]

Homework Statement

Suppose there are 1000 marbles/meter^3

And I have a cylinder of \pir^2(h), where radius is 2 meters and height is 5 meters.

How many marbles can the cylinder hold?

Homework Equations

Do I have to somehow convert volume of the form \pir^2(h) to volume of the form meters^3? If so, how can this be done? For example, if I have a cylinder of dimensions \pi3^2(10), how can I describe this volume in terms of meters^3?

The Attempt at a Solution

This is not an assigned homework question, just self study.

This is a very difficult and non-trivial problem; only recently have some problems of this type been solved. The difficulty is that volumes alone do not tell the whole story. You can have two different shaped containers having the same volume, but one can hold a significantly different number of marbles than the other. You have to pack the marbles together, so how the marbles can be made to fit partially into the spaces between other marbles will matter a lot.

For more material on this, Google "sphere-packing problem" and obtain numerous links, such as
http://en.wikipedia.org/wiki/Sphere_packing or
http://mathworld.wolfram.com/SpherePacking.html or
http://www.maa.org/devlin/devlin_9_98.html
Other links with much longer url's give pdf files of some of the recent research on the issue; they are all difficult to read and involve advanced mathematics.

RGV

 

[LearninDaMath]

This is a very difficult and non-trivial problem; only recently have some problems of this type been solved. The difficulty is that volumes alone do not tell the whole story. You can have two different shaped containers having the same volume, but one can hold a significantly different number of marbles than the other. You have to pack the marbles together, so how the marbles can be made to fit partially into the spaces between other marbles will matter a lot.

For more material on this, Google "sphere-packing problem" and obtain numerous links, such as
http://en.wikipedia.org/wiki/Sphere_packing or
http://mathworld.wolfram.com/SpherePacking.html or
http://www.maa.org/devlin/devlin_9_98.html
Other links with much longer url's give pdf files of some of the recent research on the issue; they are all difficult to read and involve advanced mathematics.

RGV

Okay thanks Ray, so given that a problem of this kind is very difficult and ignoring any attempt at finding the number of marbles that can fit in the cylinder, what if I just want to know:

How many cubic meters are exist in a cylinder of a given dimension? For instance, if I have a cylinder of πr^2(h) when r=3 and h=10, I get 282.74... but what does that actually mean? What unit of volume is the number 282.74? Is it 282.74 m^3? Or is it some other unit that I might be unaware of? Do I need to do some further converting in order to represent it in terms of meters^3?

 

[Mark44]

Okay thanks Ray, so given that a problem of this kind is very difficult and ignoring any attempt at finding the number of marbles that can fit in the cylinder, what if I just want to know:

How many cubic meters are exist in a cylinder of a given dimension? For instance, if I have a cylinder of πr^2(h) when r=3 and h=10, I get 282.74... but what does that actually mean? What unit of volume is the number 282.74? Is it 282.74 m^3? Or is it some other unit that I might be unaware of? Do I need to do some further converting in order to represent it in terms of meters^3?

The units of volume here would be cubic meters, or m³. There is also a somewhat obscure unit in the metric system - a stere. A stere is one cubic meter.

 

[Borek]

There is also a somewhat obscure unit in the metric system - a stere. A stere is one cubic meter.

It is so obscure that's the first time I hear about it

 

[tainted]

Okay thanks Ray, so given that a problem of this kind is very difficult and ignoring any attempt at finding the number of marbles that can fit in the cylinder, what if I just want to know:

How many cubic meters are exist in a cylinder of a given dimension? For instance, if I have a cylinder of πr^2(h) when r=3 and h=10, I get 282.74... but what does that actually mean? What unit of volume is the number 282.74? Is it 282.74 m^3? Or is it some other unit that I might be unaware of? Do I need to do some further converting in order to represent it in terms of meters^3?

Well what units are "r" and "h" in?

 

[LCKurtz]

The units of volume here would be cubic meters, or m³. There is also a somewhat obscure unit in the metric system - a stere. A stere is one cubic meter.

Out with cubic centimeters! In with nanosteres!

 

[Mark44]

It is so obscure that's the first time I hear about it

Gee, I thought you guys in Europe would be well-versed in all things metric. I'm shocked!

Give me the good ol' English system, with such handy units as furlongs/fortnight, etc.

 

[Mark44]

Out with cubic centimeters! In with nanosteres!

Love it!

While we're on the subject of obscure SI units, there are also steradians... (http://en.wikipedia.org/wiki/Steradian)

 

[Borek]

Cubic meters is something I hear about (and use) quite often, but stere... In Poland we sometimes call it "kubik" or "metr kubiczny". Funny thing is "kubik" (noun) and "kubiczny" (adjective) are obvious copies of "cube" and "cubic" - but they are not used in Polish in other contexts. In general "cubic" is "sześcienny" (adjective, made from "sześcian" - a cube).

At the same time steradian I know.

 

[LearninDaMath]

thanks for the feedback and i enjoyed reading the rest of the light hearted dialogue. appreciate your help.