Spheres inside a cylinder

[mathsupanova]

Hello!

Have stumbled upon a tricky problem and I don't know what to do!

the problem is as such: I have a tube (cylinder) of dimensions: height - 90cm and diameter 15cm. I have an unlimited number of spheres of diameter 2cm. - my question is: how many spheres can I fit into the cylinder?

So I have worked out the volume of the cylinder and the volume of each sphere and then divided the cylinder's volume by the volume of the sphere to come up with a number which is ridiculously too large - like 3500 spheres.
I understand that there are gaps between the spheres as they are spherical (!) but can't work it out any further. I have asked a few friends to try this out and kind of guestimate and we all end up with answers in the 1200-1500 zone, although the lack of accuracy is frustrating! Any help much appreciated.

 

[stapel]

I have a tube (cylinder) of dimensions: height h = 90cm and diameter d = 15cm. I have an unlimited number of spheres of diameter 2cm. My question is: How many spheres can I fit into the cylinder?

So I have worked out the volume of the cylinder and the volume of each sphere and then divided the cylinder's volume by the volume of the sphere to come up with a number which is ridiculously too large - like 3500 spheres.

This assumes that the spheres are malleable (like Play-Do, maybe) and will contort to take up all space, but are not compressible (like sponges) so they'll not lose any volume. Are these parameters specified in the exercise?

I understand that there are gaps between the spheres as they are spherical (!) but can't work it out any further. I have asked a few friends to try this out and kind of guestimate and we all end up with answers in the 1200-1500 zone, although the lack of accuracy is frustrating!

What sort of "packing" formula did they give you (in class, in the textbook, in the instructions to the assignment, or wherever)? You may find the topic of "sphere-packing" to be rather complex (for instance, here, here, or here), so we'll need this contextual information before proceding.

When you reply, please include a clear listing of your thoughts and efforts so far. Thank you!

 

[mathsupanova]

Thanks for your response!

to be honest I only did the volumes of the spheres and the cylinder and divided them - had no
idea where to go from there... I can now see why, as it is completely beyond my level of mathematical
ability so I might just stop there.

Thanks again!

 

[mathsupanova]

Would you believe I totally tried this with a handful of leftover Bonbons and a mug?!!

Basically I filled it and counted how many balls were in there, then I calculated the difference between this and the predicted out-put of the formula of dividing volume of cylinder minus volume of each individual sphere. Next I multiplied by 0.64 (which was a rule of thumb suggested for people when trying to guess the amount of sweets in a jar - to account for spaces between sweets). This was still over-estimating the amount of bonbons I could fit in the mug so I worked out the difference between the calculation estimate and the actual amount and used this as a constant (ended up being to multiply by about 0.55). Hey presto! There was a number within the range of guesstimates/calculations all my friends had come up with!

Is this Nobel Prize worthy or what!!? ;P

(ps: I didn't even pass high school math!)