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The big crunch - a problem for the biologists



I will teach the second semester of the "noncalculus" physics course here
at SFU starting in January. This course is intended for, among others,
biological sciences students. Among the topics that need to be covered
are radioactive decay and RC problems*. I've been using the CAPA** system
for my students' problem set assignments. Each term I familiarize my
students with the system by giving them an initial problem set which does
not count for credit, but which puts them through the mechanics of using
the system. I will use the following problem (which I made up many years
ago and called "The Malthusian Nightmare") on that set. I used it last
year and half a dozen of the 200 students got it right, including solving
(numerically) the transcendental equation that results! I take this
opportunity to demonstrate that even modest growth is not "sustainable".
This problem does so, and with a vengence! I will send the solution (in
..pdf format) to anyone who requests it.

The Big Crunch

The following problem is given to you as a challenge. It illustrates the
dominance of the exponential function over a power law. We will encounter
exponential growth (or decay) again later in the course.

Imagine an infinite three dimensional sea of nutrient. In this sea we
place one bacterium of volume a^3. This bacterium metabolizes, doubles its
volume, and divides into two new bacteria identical to the original in
time T. During the intervening time it can swim with a maximum speed v.

If this colony does not expand by swimming the bacteria will soon run out
of nutrient. Worse still, they will fill the volume of space available to
them. If the bacteria swim sufficiently cleverly they can occupy a volume
of the sea which increases in time. Eventually, however, their exponential
growth will catch up with their expansionist strategy and they will run
out of nutrient and completely fill all available space.

It is clear that the optimal strategy will involve filling a sphere of
maximum size attainable under the circumstances given.

1. Given a =3D 50 =B5m, T =3D 1200 s, and v =3D 2.0 mm/s, calculate how long=
the
colony can survive.

2. Given these bacteria, but supposing they can swim with v =3D 3.00E7 m/s
(one tenth the speed of light), what is the radius of the colony on
doomsday? Express your answer in astronomical units (AU), the distance
of the Earth from the Sun. 1 AU =3D 1.50E7 km. Before you calculate your
answer try to guess whether it will be more nearly the size of the
Earth, the solar system, the galaxy, or the universe.

Leigh

* I use a simple differential equation in presenting both of these topics.
Our Biological Sciences Department has asked us to include some calculus
in the course their students take. They want their students to have some
facility with calculus when they get to junior and senior courses, and
their students also must take a watered down year long calculus sequence
which is a corequisite to the physics course.

**http://www.pa.msu.edu/educ/CAPA/