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a. A traditional problem in the mathematics of exponentials is the following:

Since the amount on the first square is 1, on the second is 2, the third
is 2^{2}, etc., since there are 64 squares on the chessboard, the
final square gets 2^{63} grains (and the total is 2^{64} -1).
To get a sense for how much things grow after various doublings, evaluate
the following in scientific notation:

- amount after 5 doublings: 2
^{5}= - amount after 10 doublings: 2
^{10}= - amount after 20 doublings: 2
^{20}= - amount after 50 doublings: 2
^{50}= - amount after 64 doublings: 2
^{64}=

b. An streptococcus bacterium can reproduce itself in about 30 minutes if there is adequate nutritional materials and appropriate conditions. If one bacterium gets in your system and you have no immune mechanism to destroy them or limit their growth, how many would there be in day if they all reproduced freely without restraint? Assuming they are using the materials in your body to build themselves, estimate what fraction of your body mass they would have consumed in one day. (You will need to find the approximate mass of a streptococcus bacterium.)

c. The population of the world is currently growing by about 1% per year. How many years will it take for the population of the world to double? (If you don't remember how to do exponentials, do it by hand.)

* Taken from http://classes.yale.edu/fractals/Chaos/Doubling/Doubling.html
(but this is a standard problem).** **

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Page last modified February 6, 2008: G30