Expontial Models

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There are times when we want to understand growth that is not constant. An example of this would be the growth of a virus. As time goes by the virus growth rate increases more and more. Another example would be in the world of finance when we are dealing with interest.

In situations like the ones mentioned above, it is critical to understand the use and application of exponential models. This post will go through examples of the use of exponential models.

Finance Example

One common exponential model in finance is for compounded interest. The equation is as follows…

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Below is a simple word problem that calls for this equation

You invest $10,000 in a mutual fund to prepare for retirement. The interest rate is 5% compounded monthly, how much will be in the account when you plan to retire in 25 years. 

Below is what we now

  • balance = ?
  • principal = $10,000
  • rate = 0.05
  • years=  25
  • times in year = 12 * 25 = 300

Now, we simply plug this information into the equatiom to get the answer.

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The answer is shown above. The initial investment would grow to almost $35,000 dollars over 25 years.

Continuous Growth

In some fields, such as the life sciences, you want to now the growth of a virus or bacteria. Unlike in finance where the balance grows several times a year,  a bacteria is growing continuously. This leads to a slightly different exponential model as shown below.

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e is an irrational number that serves as the base. With this information, we can address the problem below

A student starts their experiment with 10 bacteria. He knows the bacteria grow 100% every hour. He will come back and check in 12 hours. How many bacteria will he find?

Here is what we know

  • final size =?
  • initial size = 10
  • rate = 1/hour
  • time = 12

We plug this into the equation to get the answer

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As you can see, the growth of the bacteria is almost incomprehensible in such a short time. This is the power of exponential growth.

Conclusion

Exponential models provide another way to find answers to questions people have. Whether the growth is over a certain number of times or continuously the model can be adjusted to deal with either of this situations.

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