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### Population Growth Rate

```
Date: 9/5/96 at 12:52:58
From: Anonymous
Subject: Population Growth Rate

If a population increases from 10 million to 300 million in 10,000
years, what is the annual growth rate?

Please send us the formula used to solve the equation.

Thanks,
Charlie
```

```
Date: 9/5/96 at 13:56:20
From: Doctor Tom
Subject: Re: Population Growth Rate

Assuming that the population grows at a constant rate (a VERY bad
assumption, I might add), then if the rate is r, every year there
will be (1+r) times as many individuals in the population.

For example, if the growth rate is 1 percent, every year there are
1.01 times as many individuals.  So after 2 years, there will be
(1+r)*(1+r) times as many, and so on.

In 10000 years, there will be (1+r)^10000
((1+r) multiplied by itself 10000 times) times as many individuals.

So the equation to solve is this:

10000000*(1+r)^10000 = 300000000

or, dividing by 10 million:

(1+r)^10000 = 30

Take logarithms of both sides:

10000*log(1+r) = log(30)

I'll use log base 10 (it doesn't matter):

10000*log(1+r) = 1.47712125472

or

log(1+r) = .000147712125472

or, taking anti-logs:

1+r = 1.00034017759

So the average annual growth rate is .034 percent --
34 thousandths of a percent -- very tiny.

-Doctor Tom,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
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