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If they say the population growth rate is "x% *per year*", then, yeah :) Hell,
for bacteria, it might easily be 1% per minute, given a rich environment.
Of course you're right: each of the expressions are approximations. I gave
the "70 rule-of thumb". 72 is also commonly used, because it has several
integral divisors. Once one starts quibbling about using 69 instead of 70, one
might as well use the _proper_ expressions. Too, the error introduced with the
approximations is quite manageable, and is usually lost in the noise of mutation
rates, environmental constraints, and so on (for "simple" populations of
bacteria or fish, say), and absolutely swamped by things like migrations and
societal changes (for humans, say).
/**************************************
"The four points of the compass be logic, knowledge, wisdom and the unknown.
Some do bow in that final direction. Others advance upon it. To bow before the
one is to lose sight of the three. I may submit to the unknown, but never to the
unknowable." ~~Roger Zelazny, in "Lord of Light"
***************************************/
________________________________
From: Robert Cohen<Robert.Cohen@po-box.esu.edu>
To: Forum for Physics Educators<phys-l@carnot.physics.buffalo.edu>
Sent: Wed, September 22, 2010 4:27:24 AM
Subject: Re: [Phys-l] H. Sapiens
When someone says the population growth rate is x%, how much larger is
the population after one year? Wouldn't it be x%? In other words, how
long does it take for the population to equal 100%+x% of what it was
originally? The various expressions give different results.
----------------------------------------------------------
Robert A. Cohen, Department of Physics, East Stroudsburg University
570.422.3428 rcohen@po-box.esu.edu http://www.esu.edu/~bbq
From: curtis osterhoudt<flutzpah@yahoo.com>
Date: Thu, 16 Sep 2010 10:22:43 -0700 (PDT)
Remember the simple rule-of-thumb: If something is growing at x% per
time y,
the doubling time is roughly 70/x to double in y units. That is, a
percentage
growth rate of (say) 1% leads to a doubling of population in about 70
years. That's
_scary_ to me, for _any_ population.
-----Original Message-----_______________________________________________
From: phys-l-bounces@carnot.physics.buffalo.edu
[mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf
Of Ken Caviness
Sent: Tuesday, September 21, 2010 8:20 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] H. Sapiens
ln2/r gives the doubling time for continuously compounded
interest and exponential growth.
ln2/ln(1+r) gives the doubling time for interest compounded
annually. [Check by solving 2A = A(1+r)^t for t.] The
slightly larger resulting time is to be expected.
ln2/(n ln(1+r/n) gives the doubling time for interest (rate
r) compounded n times per year. [Check by solving 2A =
A(1+r/n)^(n t) for t.] For an interest rate of 5.1% = .051,
compounded monthly (n=12), this actually gives 13.62 years.
If one is content with sufficiently approximate results, the
"rule of 70" can be thought of as a rough approximation of
any of these. It's systematically high as an approximation
for true exponential growth (but getting better for larger
r), for the others the error varies depending on r and n,
crossing from negative to positive near 2% for annually
compounded interest, and somewhere around 25% for monthly
compounded interest. For interest rates between 2% and 25%
one can assume that the rule of 70 will give answers that are
too small for annual interest, too large for monthly
interest; it is always too large for continuously compounded
interest (exponential growth).
I just graphed the difference between the correct formulas
and the rule of 70 approximation for the 3 cases:
exponential, annual and monthly, the pdf generated is here:
http://www.southern.edu/~caviness/RuleOf70Errors.pdf
Basically this is the following Mathematica command:
Plot[Evaluate[{Log[2]/r , Log[2]/Log[1 + r], Log[2]/(12 Log[1
+ r/12])} - 70/(100 r)], {r, .01, .5}]
The original comment related to exponential growth, not
finance at all, so the "Pe^(rt)" formula holds, and ln2/r is
the most appropriate formula for doubling time. As an
approximation for this, the rule of 70 gives answers that are
too large and really doesn't save much in the way of
calculator button clicks. It may be easier to remember, but
thinking of "doubling" might jog the memory to remember the "ln2".
Ken
-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu
[mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf
Of Robert Cohen
Sent: Tuesday, 21 September 2010 6:10 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] H. Sapiens
It seems to me that the ln2/r rule is a little worse than the
70/r rule.
Using ln2/ln(1+r), I get ln2/ln(1+.051) ~ 13.9 years.
----------------------------------------------------------
Robert A. Cohen, Department of Physics, East Stroudsburg University
570.422.3428 rcohen@po-box.esu.edu http://www.esu.edu/~bbq
-----Original Message-----business or
From: phys-l-bounces@carnot.physics.buffalo.edu
[mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of Ken
Caviness
Sent: Thursday, September 16, 2010 8:57 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] H. Sapiens
Ah, the "Rule of 70". That's sometimes taught in finance,
economics classes. I'm not sure why. Most people can'tdivide x into
70 without a calculator anyway, and most calculators have an "ln x"extensibility
button, so it would be basically as quick to use the more exact
formula, ln2/r. And this has the advantage of immediate
to questions such as, "How long does it take for a populationcompounded
increasing at rate x to triple?" -- (ln3/r)
This is just exponential growth, also known as continuously
interest, which I was taught to remember using the "pert"formula: A
= P e^(rt), where P is the principal or originalnumber/amount, A is
the future amount, t is the time in some convenient units, r is thesometimes using
interest or growth rate (per time unit). The more standard
exponential growth/decay formula is N = N_0 e^(+-r t),
lambda instead of rate r, but it's the same thing.= ~ 13.6
To get the rule of 70, I just plugged in P = 1, A = 2:
A = P e^(rt) ==> 2 = e^(rt) ==> ln2 = rt ==> t = ln2/r ==> t = ~
0.693/r = 69.3/(100r) = ~70/x.
Of course, you have to use the decimal for the rate instead of the
percentage. :-)
So doubling time with a rate of 5.1% per year is ln2 / .051
years. Using 70/5.1 gives ~ 13.7 years, so the rule of 70is not bad,
just (in my opinion) unnecessary.Behalf Of curtis
Cheers,
Ken
-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu
[mailto:phys-l-bounces@carnot.physics.buffalo.edu] On
osterhoudtat x% per
Sent: Thursday, 16 September 2010 1:23 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] H. Sapiens
Remember the simple rule-of-thumb: If something is growing
time y, the doubling time is roughly 70/x to double in yunits. That
is, a percentage growth rate of (say) 1% leads to a doubling ofsubmit to
population in about 70 years. That's _scary_ to me, for _any_
population.
/**************************************
"The four points of the compass be logic, knowledge, wisdom and the
unknown.
Some do bow in that final direction. Others advance upon it.
To bow before the one is to lose sight of the three. I may
the unknown, but never to the unknowable." ~~Roger Zelazny,in "Lord
of Light"five years
***************************************/
________________________________
From: Bernard Cleyet<bernardcleyet@redshift.com>
To: Forum for Physics Educators<phys-l@carnot.physics.buffalo.edu>
Sent: Thu, September 16, 2010 11:15:34 AM
Subject: Re: [Phys-l] H. Sapiens
Contrary to what I presume many think, Mexico is less adding to that
problem:
"Throughout most of the twentieth century Mexico's population was
characterized by rapid growth. Even though this tendency has been
reverted and average annual population growth over the last
was less than 1%, the demographic transition is still inprogress, and
Mexico still has a large cohort of youths."optimistic about
http://en.wikipedia.org/wiki/Demographics_of_Mexico
bc
Demographic transition:
http://anthrocivitas.net/forum/showthread.php?t=1539
On 2010, Sep 11, , at 20:15, ludwik kowalski wrote:
Actually, I was wrong about the "less than 20 years, as youcan see at:
http://en.wikipedia.org/wiki/World_populationwill probably
Ludwik
= = = = = = = = = = = = = = = = = = = = = = = = = = =
On Sep 11, 2010, at 11:05 PM, ludwik kowalski wrote:
On Sep 11, 2010, at 10:49 PM, brian whatcott wrote:
There are less than 8 billion people presently on Earth. . . .This is about four times more than when I was a kid. It
double again in less than 20 years. How can one be
______________________________________________________________________________________________the future of sapients?Ludwik
Ludwik
http://csam.montclair.edu/~kowalski/life/intro.html
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