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### Doubling Bacteria

Date: 09/17/97 at 00:42:57
From: Russell
Subject: Exponents

Dr. Math,

"A biologist notices that a certain bacterium splits into 2 separate
bacteria once every 15 minutes.  If there was one bacterium on the
side 3 hours ago, how many are there on the slide now"?

I am in the sixth grade.

Date: 10/04/97 at 15:19:30
From: Doctor Chita
Subject: Re: Exponents

Hi Russell:

This is a very interesting problem for a sixth grader.

One way to solve it is to make a table with two columns. Label the
first column "time (minutes)" and the second column "number of
bacteria."

It would look like this:

time (minutes) | no. bacteria
---------------------------------
0         |  1  (At the start there was only 1 bacterium.)
|
15         |  2  (This is the way these bacteria reproduce.)
|
30         |  4  (Each bacterium split in 2 during this time.)
|
45         |  8

and so on.

Continue the table until you get to 3 hours. (Don't forget to change
hours to minutes.)

You should see a pattern when you're done. You can then use this
pattern to figure out how many bacteria there would be in 10 hours or
100 hours. Quite a lot!

If you know some algebra, you can also solve this formula using an
exponential equation:

n = k*2^(t/d)

where n is the number of bacteria after t minutes, k is the number
of bacteria at time zero, d is the time in minutes that it takes to
double, and t is the time in minutes at which we are counting the
bacteria. The "^" means that what follows is an exponent.

The formula may look a little strange, but if you think about it a
bit you can see why it makes sense. First of all, the number of
bacteria in the sample at any time depends on how they reproduce and
how long they reproduce, as well as how many there were to start
with.

The 2 in the equation represents the fact that each time a bacterium
reproduces, it splits in two halves.

The exponent, (t/d), represents how many times the bacteria have
doubled. The numerator of the fraction is the time at any moment
(in the problem, it's 3 hours). The denominator of the fraction
represents the fact that the bacteria split every d minutes (15 in
the problem). Therefore, the number of "splitting times" is the
total time divided by the length of one doubling period. That is, 3
hrs/15 minutes, or in minutes, 180/15 periods. It will have doubled
12 times in those 3 hours.

The constant k in the equation is the number of bacteria you start
with, since at time t=0 the exponent is 0, and 2^0 is 1. In this
case, it is just 1 -- the one bacterium you start with.

Now, change time from hours to minutes (3 hours = 180 minutes). Then
substitute the values of k, t, and d into the right side of the
equaton and solve for n.

n = 1*2^(180/15)

n = 2^(12)

The equation says that the number of bacteria depends on each
bacterium dividing in 2 at the end of twelve 15-minute periods.
You may need a calculator to simplify 2^(12).

Check that this answer is the same as the one you found using a table.
Can you see the pattern in this solution? (You might want to think
about how many bacteria you would have after 3 hours if each bacterium
divided into 3 parts instead of 2 each time. Make a table and then use

Anyway, I hope these bacteria are the good kind! The biologist will
have lots of them in his sample.

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

Associated Topics:
Middle School Exponents
Middle School Word Problems

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