MHB How many marbles are in a jar?

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The discussion centers around estimating the number of marbles in a jar, with calculations suggesting there should be 262 marbles based on the jar's volume and the volume of individual marbles. However, the actual count is 167 marbles, indicating a discrepancy. The difference arises from the assumption that the jar is 74% filled, which does not account for the irregular packing of the marbles. The calculations highlight the importance of considering the space between marbles when estimating quantities. The issue has been identified and resolved by the original poster.
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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?
 
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I have figured out the problem. Thank you.
 

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