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Order of Magnitude Calculations

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Legend has it that, many centuries ago, Archimedes jumped out of his bathtub and ran across town naked screaming "Eureka!" after he solved an especially difficult problem. Though you may not have thought of things this way before, when you drink a glass of water, the water that you are drinking contains some water molecules that were in Archimedes' bathwater that day, because water doesn't get created or destroyed on a large scale. It follows the water cycle, which includes rain, evaporation, flowing of rivers into the ocean, and so on. In the more than two thousand years since his discovery, the water molecules from Archimedes' bathwater have been through this cycle enough times that they are probably about evenly distributed throughout all the water on the earth. When you buy a can of soda, about how many molecules from that famous bathtub of Archimedes are there in that can?

Round the answer to the nearest power of 10 and then express your answer as the order of magnitude. For instance, if your estimated answer is 3 times 10^5, enter 5. If your estimated answer is 8.7 times 10^5, you should enter 6 (rounding up to the next power of 10).

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Suppose that the water that was in Archimedes' bathwater is indeed evenly distributed throughout all the water on the earth. If you denote by N the total number of water molecules on earth, and by N_arch the number of water molecules that were in Archimedes' bathwater, then the fraction of water molecules in some arbitrary sample of water that had been in Archimedes' bath is:

f = N_arch/N

If you multiply the numerator and the denominator by the mass of a single ...