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 July 8th, 2012, 08:16 AM #1 Senior Member   Joined: Jan 2010 Posts: 205 Thanks: 0 How many balls can fit into a jar The jar has a radius of 6" and a height of 24" and each ball has a radius of 1". So I found the volume of the jar which is $\pi6^{2}(24)= \approx 2,714.33605$ and the volume of the balls which is $\frac{4}{3}\pi1^{3}= \approx 4.1887902$ And then I divided how many of the balls can go into the jar by dividing: $2714.33605 \div 4.1887902= 648 balls$ Does that number take into account the spaces between the balls when put into the jar? Like the small gaps when spheres are placed next to each other.
 July 8th, 2012, 09:12 AM #2 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,328 Thanks: 1024 Re: How many balls can fit into a jar 648 is correct IF each ball is melted into a liquid. Before you go nuts with this, google "packing identical spheres in a cylinder"...
July 8th, 2012, 09:47 AM   #3
Math Team

Joined: Dec 2006
From: Lexington, MA

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Re: How many balls can fit into a jar

Hello, daigo!

Quote:
 The jar has a radius of 6" and a height of 24", and each ball has a radius of 1". How many balls can fit into the jar? $\text{So I found the volume of the jar which is: }\:\pi(6^2)(24)\:=\: 864\pi \;\;\;\text{and the volume of a ball which is: }\:\frac{4}{3}\pi(1^{3}) \:=\:\frac{4}{3}\pi$ And then I found how many of the balls can go into the jar by dividing: [color=beige]. . . [/color]$864\pi \,\div\,\frac{4}{3}\pi \:=\: 648\text{ balls.}$ Does that number take into account the spaces between the balls when put into the jar? Like the small gaps when spheres are placed next to each other?[color=beige] . [/color] [color=blue] . . . No[/color]

You have melted 648 balls.
[color=beige]. . [/color]Then you poured the liquid into the jar.
You have filled the jar with the equivalent volume of 648 balls.

If we leave the balls intact, the problem becomes difficult.

We can arrange the balls like this on the floor of the jar.

[color=beige]. . [/color]$\begin{array}{c} \circ \\ [-2mm] \,\circ\;\circ\ \\ [-3mm] \circ \\ [-2mm] \circ\;\circ \\ [-2mm] \circ \\ \end{array}$

Then we can place 6 more balls around them
[color=beige]. . [/color]and have a large hexagon of 13 balls.
These will fit inside the jar.

These layers are 2 inches high.
We can fit 12 such layers in the jar.

Hence, $12\,\times\,13 \:=\;156\text{ balls.}$

$\text{If we turn each layer }30^o\text{ to the layer below it,}
\;\;\text{more layers can be fitted into the jar.}$

I'll let you work out the details . . .

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# how to calculate how many balls in a jar

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