Exponential Growth in Dividing Bacteria

Date: 01/08/2004 at 18:11:14
From: Jessica
Subject: dividing bacteria cells

Suppose a bacterium divides in half an hour, then the two resulting
bacteria divide in the next half an hour.  At this rate, how many
bacteria would there be in 72 hours?

Date: 01/09/2004 at 08:03:40
From: Doctor Jason
Subject: Re: dividing bacteria cells

Hi Jessica,

Thanks for writing to Dr. Math!

If we start with one bacterium, then 30 minutes later it has divided, 
and we have 2.  After 30 more minutes, each of those bacteria have 
divided, and we have 4 bacteria.  Thirty more minutes later we have 
8, then 16, and so on.

It is important to realize that though the bacteria are dividing, the 
number of bacteria we have is doubling, or being multiplied by two.

Let's look at the data for the first few hours and organize it in a 
table.  

   Time Pd. | Time Elapsed | # of bacteria
   -------------------------------------------------
       0    |  0.0 hrs     |      1
       1    |  0.5 hrs     |      2  (2)
       2    |  1.0 hrs     |      4  (2 * 2)  
       3    |  1.5 hrs     |      8  (2 * 2 * 2)
       4    |  2.0 hrs     |     16  (2 * 2 * 2 * 2)
       5    |  2.5 hrs     |     32  (2 * 2 * 2 * 2 * 2)
       6    |  3.0 hrs     |     64  (2 * 2 * 2 * 2 * 2 * 2)

Notice that the Time Period in the above table corresponds to the 
number of 2's being multiplied together.  Since we are only working 
with the number 2, the size of our bacteria population can be
expressed as a power of two.  For example, 2^2 equals 4.  Let's edit 
the data table to show this:

   Time Pd. | Time Elapsed | # of bacteria
   -------------------------------------------------
       0    |  0.0 hrs     |      1 = 2^0
       1    |  0.5 hrs     |      2 = 2^1 
       2    |  1.0 hrs     |      4 = 2^2
       3    |  1.5 hrs     |      8 = 2^3
       4    |  2.0 hrs     |     16 = 2^4
       5    |  2.5 hrs     |     32 = 2^5
       6    |  3.0 hrs     |     64 = 2^6

By analyzing the table we find that we can determine the number of 
bacteria for a given time period by raising two to the power of the 
number that represents the time period (2^Time Period).  If we wanted 
to know how many bacteria we would have after 24 hours, we would 
first need to know what time period we are working with.  Since there 
are two 30-minute time segments per hour, after 24 hours, there would 
be 48 cell divisions (Time Elapsed in hours * 2 = Time Period).  At
the end of 24 hours, there would be 2^48 or 2.81474977 × 10^14 bacteria.

The general rule for this problem is:

     number of bacteria = 2^(Time Period)  -or-

                        = 2^(Time Elapsed in hours * 2)

This sort of problem illustrates an idea called exponential growth,
and one characteristic of such problems is that the growth starts out
fairly slowly, but then begins to really accelerate.  Can you see that
happening in our data?  Any exponential growth problem involves
multiplying some starting amount repeatedly by a number greater than 1.

So, how many bacteria will there be after 72 hours?  Let me know what
you think.  I think you'll need a calculator for this one!

I hope this helps! Let me know if there is anything else I can do.

- Doctor Jason, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/09/2004 at 12:02:23
From: Jessica
Subject: Thank you (dividing bacteria cells)

Thank you very much for helping me solve the bacteria question!