Doubling Bacteria

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

Dr. Math,

Help me with this please...

"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 
the formula to check your answer.)

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/