HW3 Population Growth Solution
Content
1 of 3
Teaching Physics with the Physics Suite
Edward F. Redish
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?"*
2
Since the amount on the first square is 1, on the second is 2, the third is 2 , etc.,
63
since there are 64 squares on the chessboard, the final square gets 2
grains (and
64
the total is 2 -1). To get a sense for how much things grow after various doublings,
evaluate the following in scientific notation:
5
amount after 5 doublings: 2 =
10
=
20
=
50
=
64
=
amount after 10 doublings: 2
amount after 20 doublings: 2
amount after 50 doublings: 2
amount after 64 doublings: 2
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? (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
2 of 3
is a standard problem).
a.
5
1
2 = 32 = 3.2 x 10 ;
10
= 1024 = 1.0 x 10
20
= 1.0 x 10
6
50
= 1.1 x 10
15
64
= 1.8 x 10
19
2
2
2
2
3
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
15
14
doublings divided by 4, so we have about 0.3 x 10 = 3 x 10 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)
3
3
-18
V = (4/3)πr ~ 4(d/2) = 4/8 x 10
3
-18
3
m = 0.5 x 10
m .
Since the bacillus travels in water, is should have about the same density as water,
3
1000 kg/m . So a strep bacillus should have a mass about
3
3
-18
m = ρV = (10 kg/m )(0.5 x 10
3
-15
m = 0.5 x 10
14
So the total mass of strep after one day would be 3 x 10
kg.
-15
x 0.5 x 10
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
3 of 3
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
N
of (1.01) . To find for what N you get a doubling, you need to solve.
N
(1.01) = 2.
We do this by taking the logarithm of each side.
N log(1.01) = log 2
so
-3
N = (log 2)/(log 1.01) = 0.301/(4.32 x 10 ) = 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.)
Page last modified February 9, 2008: G30
Teaching Physics with the Physics Suite
Edward F. Redish
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?"*
2
Since the amount on the first square is 1, on the second is 2, the third is 2 , etc.,
63
since there are 64 squares on the chessboard, the final square gets 2
grains (and
64
the total is 2 -1). To get a sense for how much things grow after various doublings,
evaluate the following in scientific notation:
5
amount after 5 doublings: 2 =
10
=
20
=
50
=
64
=
amount after 10 doublings: 2
amount after 20 doublings: 2
amount after 50 doublings: 2
amount after 64 doublings: 2
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? (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
2 of 3
is a standard problem).
a.
5
1
2 = 32 = 3.2 x 10 ;
10
= 1024 = 1.0 x 10
20
= 1.0 x 10
6
50
= 1.1 x 10
15
64
= 1.8 x 10
19
2
2
2
2
3
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
15
14
doublings divided by 4, so we have about 0.3 x 10 = 3 x 10 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)
3
3
-18
V = (4/3)πr ~ 4(d/2) = 4/8 x 10
3
-18
3
m = 0.5 x 10
m .
Since the bacillus travels in water, is should have about the same density as water,
3
1000 kg/m . So a strep bacillus should have a mass about
3
3
-18
m = ρV = (10 kg/m )(0.5 x 10
3
-15
m = 0.5 x 10
14
So the total mass of strep after one day would be 3 x 10
kg.
-15
x 0.5 x 10
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
3 of 3
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
N
of (1.01) . To find for what N you get a doubling, you need to solve.
N
(1.01) = 2.
We do this by taking the logarithm of each side.
N log(1.01) = log 2
so
-3
N = (log 2)/(log 1.01) = 0.301/(4.32 x 10 ) = 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.)
Page last modified February 9, 2008: G30
