Introduction to Linear and Exponential Growth

Date: 11/04/2003 at 19:12:07
From: Christina
Subject: Exponentially Growth

As a biology project, Nicole is investigating how fast a particular 
beetle population will grow under controlled conditions.  She started 
her experiment with 5 beetles.  The next month she counted 15 beetles. 

If the beetle population is growing linearly, how many beetles can 
Nicole expect to find after 2, 3, and 4 months?

If the beetle population is growing exponentially, how many beetles 
can Nicole expect to find after 2, 3, and 4 months?

For the first question, I think that you only have to double the 
number of beetles from the last month that would equal to 30, and 
then 60.  But I am not sure if I am right or not.  And the second
question, I don't know how to figure out.

Date: 11/04/2003 at 23:40:34
From: Doctor Riz
Subject: Re: Exponentially Growth

Hi Christina -

Thanks for writing and asking a good question.  As you have realized, 
linear and exponential growth are not at all the same thing.  Let's 
take a quick look at each one and how it applies here.

In linear growth, the amount of change for a given unit of time is 
always the same.  The hair on your head grows linearly--each day it 
gets longer by the same small amount.  If you knew that each day it 
grew 0.04 inches, you could predict that in a week it would grow

  7 days * 0.04 inches per day = 0.28 inches in a week 

So in this problem, can you determine how many more beetles there are
after one month?  Whatever that amount is, the total number of beetles
will increase by the same number the next month and the month after
that and so on.

Now, as you might imagine, population doesn't really grow linearly.  
Since each month there are more beetles having babies, we would expect 
the population to start increasing by a bigger number each month. 
This is the idea behind exponential growth.
  
Suppose we know that a beetle population starts at 5 and doubles each 
month.  After one month there would be 5*2 or 10 beetles.  It would 
double again the next month, so now there would be 10*2 or 20.  In the 
third month it would double again so now there would be 20*2 or 40.

Let's look at those numbers more closely and write them in factored 
form:

  Starting amount    5             =  5     =  5
  After first month  5 * 2         =  5 * 2 = 10
  After second month 5 * 2 * 2     = 10 * 2 = 20
  After third month  5 * 2 * 2 * 2 = 20 * 2 = 40

Can you see the pattern taking place?  Each month we multiply by 2 
again.  We can also write those expressions using exponents:

  Starting amount    5             = 5       =  5
  After first month  5 * 2         = 5 * 2^1 = 10
  After second month 5 * 2 * 2     = 5 * 2^2 = 20
  After third month  5 * 2 * 2 * 2 = 5 * 2^3 = 40

If this pattern continues, how big will the population be after one
year?  Since there are 12 months in a year, it would be 

  5 * 2^12, which is 20,480!

The general form of an exponential equation is y = a * b^x where 'a'
is the 'intial amount' that you start with, 'b' is the multiplying
effect that occurs during each time period, and 'x' is the number of
time periods that take place.

So, if you started with 100 beetles and they tripled every two months, 
how many would there be after 1 year?  In this case

  a = 100 (starting amount)
  b = 3 (tripling effect every 2 months)
  x = 6 (there are six 2-month periods in one year)  

The population after a year would be 

  100 * 3^6 or 72,900 beetles!  

Remember that when you evaluate 100 * 3^6 the exponent calculation is
done before the multiplication.

You may be noticing that in exponential growth it tends to start
increasing slowly but then really takes off over time.

So, can you figure out what a, b, and x would be in your beetle
problem and use them to answer the questions?

Good luck!  Please write back if you are still confused and show me
what you've been able to do.

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