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Acuboid with length $10$ cm width $8$ cm and height $2\pi $ ,
What is the maximum number of spheres each with radius $1$ cm can be packed in the cuboid ?

I searched the web and found a lot of talking about this kind of problems , different and different approaches >

Why is that ? And what is the proper approache to this problem?

Please elaborate your answer so that i can understand it

Thank you for your help

Medo
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    Hint: The maximum packing density of equal spheres (https://en.wikipedia.org/wiki/Sphere_packing) is $\frac{\pi}{3 \sqrt2}$. Using this, what is the maximum number of spheres that can be packed into a box? – Toby Mak Jun 27 '17 at 10:23
  • @Toby Mak $88$? – Medo Jun 27 '17 at 10:33
  • Use that as an upper bound for your calculation. – Toby Mak Jun 27 '17 at 10:42
  • The cubic lattice (on the same page) has a density of $\pi/6$, so since there are $10 * 8 * \lfloor (2 \pi) \rfloor $ or $480$ cubes that can fit in the cuboid, then a better estimate would be $480 * (\pi * 1/6) / (4/3 * \pi)$ spheres, which is about $60$ spheres. This can be the lower bound of your calculation. – Toby Mak Jun 27 '17 at 10:48

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