# Fishy Calculation

Aaron Santos over at A Diary of
Numbers, author of How Many
Licks?,
has posted a Fermi
Contest.
For the uninitiated, a Fermi
Problem is a seemingly
unanswerable problem, which you can actually estimate reasonable by
breaking the problem down into smaller parts. They’re really fun, and I
intend to post more in the future. The question at hand is: *How far
would the oceans sink if we took all the fish out?* I’ll answer in two
very different ways.

### Preliminaries

In both cases, I’ll answer how much the ocean depth would decrease by
first calculating the total volume of fish in the ocean. Why is this
helpful? Because for a spherical shell, you can estimate its volume by
just taking its surface area and multiplying by the thickness, i.e.

### Energy Budget

First I’ll estimate the volume of fish in a rather general way. I’ll try
to do it on energy grounds. I know how much solar energy hits the earth
per meter squared on average (340 W/m^2). I’m going to assume that fish
get their energy from plankton, and plankton get their energy from the
sun, both with about 10% efficiency. I’ll also assume that life occupies
as much space as possible, probably about half of the ocean surface
counts as liveable. From this I get the total energy available to make
fish. How many fish does that allow? I’ll assume that fish use about as
much energy per kilogram as humans do. I know that humans have to take
in about 2000 food Calories or 2,000,000 calories a day to survive. From
this I get the total weight of all of the fish in the ocean, and for
their volume I assume they’re the density of water (fish are bouyant).
My calculation:

### Fishing

Next, I’ll try and estimate again, this time guessing based on how much
we fish fish. I know that overfishing is a problem, which means that for
some fish species we fish more than fish make little fishies, so if I
can estimate how much fish we eat, I can estimate how much we fish, so I
can estimate how many fish there are, and then I can estimate how much
the ocean depth changes. This calculation is very rough, I took some
very basic order of magnitude guesses at some of the parameters. In
particular, I had to guess how many fish species we fish, which I took
to be 1/100 for no very good reason. My calculation is below:

### Geometric Average

Honestly I trust my first number more than the second, but I’m going to
average my two results in order to come up with the number that I’ll
submit to the contest. But, I’m not going to average my two numbers
arithmatically: *geometrically*, i.e. I’m going to take the square root of their
product:

### My Answer

This kind of averaging is logarithmic in nature, and my experience has
been that it is much more successful average to use when you are doing
fermi problems. Doing this on my two numbers I obtain my entry:

## Comments !