A current radio commercial for an insurance company makes an interesting claim, asking the listener: "Did you every stop to pick up a penny?"
It continues: "Now, imagine doing that for each of your next 50,000 steps...that is the equivalent of what you would save if you switched to" their car insurance.
Question 1: How big of a cubic box would be needed to hold these 50,000 pennies, assuming they were stacked neatly in piles? (Feel free to convert and use "gecko-height" as your unit of length.
Question 2: Would the box have to be bigger if the pennies were thrown into the box at random? Explain.
Hint: Stack up some pennies until the height of the stack is equal to the width of a penny. Can you use this as a cubic unit...
Solution Commentary: For Question 1, the idea of a proportion seems like a good next move....
For Question 2, you might try an experiment...make a cardboard cubic box whose side length is equal to the width of 3 pennies. Fill it with neatly stacked pennies, empty it, and just throw the pennies in randomly. How much less or more volume is needed? Is it appropriate to use a proportion here?