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### Counting Bacteria with Calculus

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Date: 2/15/96 at 10:36:37
From: Daniel Peterson
Subject: Math Problem

The number of bacteria in a culture increases from 3000 to 9000
in 8 hours.  If the rate of increase is proportional to the number
present, estimate the number of bacteria at the end of 24 hours.

Daniel Peterson
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Date: 2/15/96 at 16:1:38
From: Doctor Byron
Subject: Re: Math Problem

Hi Daniel,

For answering this problem I'm going to assume you know a
little basic calculus, since that is the best way to really understand
where the answer to this type of problem comes from.

Let's call the bacteria population P(t) where t is the time.
P(0) is our initial value and is equal to 3000.  We also know that
P(8) = 9000.

Finally, we are told that the rate of increase of bacteria is
proportional to the number of bacteria currently present.  This is
the same as saying that the derivative of the population function
is equal to some constant times the population.  We therefore
have:
dP
P = k * ----  where k is an arbitrary constant.
dt

Rearranging this expression slightly, we end up with:

dt     dP
---- = ----
k      P

This now makes finding P (relatively) easy, since we can integrate
both sides.  Equations in which each variable can be isolated on a
separate side of the equation are called "separable differential
equations".  They are often much easier to solve than those in which
the variables cannot be separated.

But to continue with the problem... integrating both sides results in:

t/k = ln(P) + C

[Normally the integral of 1/P would be natural log of the absolute
value of P.  In this case, however, we know that a population must
be positive, so this distinction isn't necessary.]

Applying the exponential function to both sides (i.e., raising e to
the power of each side of the equation):

t/k    [ln(P) + C]    ln(P)     C
e   =  e            = e      *  e

(let A = e^C)

t/k
e   =  P * A

t/k
P = Be     (where B = 1/A)

Now we can use the two conditions from the problem to find the
constants B and k.

P(0) = 3000 = B(e^0) = B

Therefore B = 3000.
8/k
P(8) = 9000 = 3000e

Solving this, we find that k = 7.282 (approximately)

Now it is possible to find P(24)
P(24) = 3000e         = approx. 81,000 bacteria

You may have noticed that the three values are following a
progession proportional to the powers of 3:  3*1000, 3^2*1000,
3^4*1000.  It ispossible to show that the value at 24 hrs is
_exactly_ 81,000 bacteria.You can either do this analytically,
as above, or realize that from the problem description the
population will triple every 3 hours!  Inother words, now that
you have seen how the answer can be derived there is a very
efficient shortcut for figuring this type of problem out.  If you
know the population increases by a certain factor in a set period
of time, then it must continue to increase by that factor for each

I know it was a little mean of me to go through all those equations
when there is a quick shortcut, but learning the above method
will aid you in solving many other related problems where cutting
corners is not so easy.

-Doctor Byron,  The Math Forum

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Associated Topics:
High School Calculus

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