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Understanding Exponential Growth

Exponential Growth is used to predicte the future outcome of population growth, inflation and decay of certain waste. The rate of growth and average growth rate. 

Exponential Model/Formula


P- Predicted/Futured Population
Po- Inital Population
r- Growth Constant
t- Time

 Example:
The population in Yourtown in 1991 was 12,000, and in 2001 the population was calcuated at 20,000.  Find out the growth constant.


We plug in our known values:

Start getting our unknown on one side of the equation:

Then we use the natural log to cancel out e: LN(20,000/12,000) = 10r

Next, to get "r" (our unknown value) by itself, we can divide by sides by 10: LN(20,000/12000) / 10 = r

Solving the log function in our calculator gives us: r = 0.0510825624. This is our constant growth rate!
 
As given above, here are the step by step instructions
Step 1:  Figure out your Future or Predicted Population, Initial Population, and Time.  Plug into Forumula.
Step 2:  Next you want to get "r" alone by dividing both sides for the Inital Population.
Step3:  To get rid of the e^rt you need to take the opposite of e^ which is LN and whatever you do to one side needs to be applied to both.  That will eliminate the e^ and you are left with r*t. 
Step 4:  Next is getting the r by itself by dividing 10 by both sides.
Step 5:  Lastly is entering it in the calculator correctly.  Make sure you close your paratheses.

Once you have your Model you can now predict future outcomes!

Example 2:

Stephen purchased a home On September 1998 for $190 ,000.  In  June 1999 Calvin purchased a home close to Stephen's with the same floor plan for $208,000.  How long will it take for Stephen's home to double it's original price?

Remembering the forumla


P= Future Price
Po=Initial Price
r=Growth Constante
t=time

Find out the values and plug into the formula:


We need to put "t" in exact years
From September 1998 - June 1999 = 10 months. We divide by 12 to determine the number of years: t = 10/12.



Now, divide both sides by $190,000 to isolate the r



Next to get rid of e take the natural log function on both sides: LN(208000/190000) = r(10/12)

Get r by itself by dividing by (10/12): LN(208000/190000) / (10/12) = r

From here, we plug in the values into the calculator, and get our value for r: 0.1086160809

Now we have our exponential model for figuring out How long it will take to double its price. In this question you are looking for "t"
You want double the 189 so mulitple the inital price by 2
190 * 2 = 380
P=380

Now we have our formula we are ready to plug in again:



Next, we focus on getting "t" by itself by cancelling out e:

LN(380/190)=0.108616089t

And finally, we can get our "t" by itself:
LN(380/190)/0.108616089 = t

Using a calculator, we determine that t = 6.3. Since t was originally measured in years, t = 6.3 years. Converting 6.3 years to years and months to answer the original question, the price will double in 6 years and 4 months.

While these steps may look difficult, they really are not. It's mostly remembering to do all the steps before declaring you have a final answer!


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