How many marbles are in a jar?

[Nick C]

How many marbles are in a jar?

There are about 167 marbles in a jar.

Volume of sphere (marbles) = (4/3)πr^3
r = (4/3)π(0.25)^3
volume of sphere/marble = 0.0654 inches

Volume of cylinder (glass) = v=πr^2h
v=π(1.625)^2*2.80
volume of cylinder/glass = 23.23 inches

It is a given that the glass is 74% filled and 26% empty due to space in between the marbles.

Number of marbles = (jar volume * 0.74)/volume of one marble
= (23.23*0.74)/0.0654
= 262 marbles

Math says that there is 262 marbles when it is actually 167 marbles. Any ideas?

 

[Dr.Peterson]

Which numbers are you sure of?

I would guess that the marbles don't have exactly the radius you specified. Presumably you know enough to measure the interior dimensions of the container, but how accurate are they?

But also, your 74% is not "a given"; that is the densest possible packing, and even then it assumes no edge effects (where the container distorts the arrangement).

Suppose we guess at "random packing", which is said to take 64% of the space. Then the number will be (23.23*0.64)/0.0654 = 227. That's still more than you say, but I'm not surprised.

 

[Nick C]

You answered my question. Thank you.