2.1.4.P19

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

"According to George Gamow, chess was invented by Sissa ben Dahir, Wazir of the court of King Shiram. King Shiram loved the game so much that he offered Sissa any reward he could name. Perhaps trying to impress the king with his mathematical skills, Sissa asked for some rice, one grain on the first square of the chessboard, two on the second, four on the third, eight on the fourth, and so on, each square's amount being the double of the previous square's. How much rice did Shiram owe Sissa?"*

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} =

Estimate the mass of rice that would be on the chessboard.

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?

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

a.

2^{5} = 32 = 3.2 x 10^{1};

2^{10} = 1024 = 1.0 x 10^{3}

2^{20} = 1.0 x 10^{6}

2^{50} = 1.1 x 10^{15}

2^{64} = 1.8 x 10^{19}

So 2^{64} is the number of grains of rice on the last square. The total number will be 1 less than 2^{65} or about 4 x 10^{19} grains.

A grain of rice is approximately a cylinder of dimensions about 5 mm x 1 mm x 1 mm or 5 mm^{3 }= 5 x 10^{-9} m^{3}. The density of rice is about the same as water (since it sinks slowly when you drop it into water to cook it). Since 1 m^{3} of water has a mass of 10^{3} kg, 4 x 10^{19} grains will have a volume of

Volume = (4 x 10^{19}) x (5 x 10^{-9} m^{3}) = 2 x 10^{11} m^{3}.

so its mass is

Mass = density x volume = (10^{3} kg/m^{3}) x (2 x 10^{11} m^{3}) = 2 x 10^{14} kg.

b.

After each half hour the number of streptococci doubles. So at the end of one hour we have two doublings and at the end of 24 we have 48 doublings. This is the same as 50 doublings divided by 4, so we have about 0.3 x 10^{15} = 3 x 10^{14} strep bacilli. To see what fraction of your mass has been converted into strep, we will calculate the mass of the bacteria. (Not the volume. Mass is pretty well conserved through both physical and chemical transformations, while volume is not. Think of water freezing or turning to steam, for example.)

We need the size of a strep bacterium. Since I have no personal experience with these directly (e.g., have never had a bio class where we looked at it in a microscope), I have to look it up. The CDC Public Health Image Library includes lots of photos of it, but appalingly gives no scales on its photos. A photo of an E. coli cell in *The Molecular Biology of the Cell* (Alberts et al., p. 16), shows a cylinder 1 micrometer in diameter and 2 micrometers in length. A drawing at the top of the page suggests that a strep cell is a sphere of diameter about the same size as the diameter of E. coli. This would give the cell a volume of about (*r* is the radius, *d* is the diameter)

*V* = (4/3)π*r*^{3} ~ 4(*d*/2)^{3} = 4/8 x 10^{-18} m^{3} = 0.5 x 10^{-18} m^{3}.

Since the bacillus travels in water, is should have about the same density as water, 1000 kg/m^{3}. So a strep bacillus should have a mass about

*m* = ρ*V* = (10^{3} kg/m^{3})(0.5 x 10^{-18} m^{3} = 0.5 x 10^{-15} kg.

So the total mass of strep after one day would be 3 x 10^{14} x 0.5 x 10^{-15} kg = 0.15 kg.

This is about 0.2% of my total mass. (Note that in two days and two hours -- 4 more doublings -- it will be 3.2%, and two hours later it will be almost 50% of my total mass!)

c.

If you are familiar with logarithms, this is pretty straightforward. With each year, the population increases by a factor of 1.01. So after N years, it increases by a factor of (1.01)^{N}. To find for what N you get a doubling, you need to solve.

(1.01)^{N} = 2.

We do this by taking the logarithm of each side.

N log(1.01) = log 2

so

N = (log 2)/(log 1.01) = 0.301/(4.32 x 10^{-3}) = 70.

If you are not familiar with logarithms, you can do it by hand by multiplying 1.01 by itself many times. If I multiply it by itself 5 times I get 1.051. If I square that (multiplying 1.01 by itself effectively 10 times) I get 1.1046. If I square that (multiplying 1.01 by itself effectively 20 times) I get 1.2202. Squaring that (40 times) gives 1.4889. Squaring that (80 times) gives 2.21 -- too big. So I divide down by 1.01 until it gets to 2 -- somewhere between 69 and 70.

So at the current rate, the world population will double in 70 years.

** The energy in chemical transformations really comes from changes in the mass. Thus, when two hydrogen atoms get together and form a covalent bond the mass of the H2 molecule is slightly less than the mass of the separated atoms -- by the chemical energy divided by c2, using E = mc2. This is a very tiny fraction -- about 1 part in 10 billion.)*

Joe Redish 2/6/08

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