## Putting It Together: Growth Models

At the beginning of the module, you were introduced to some social media statistics for several celebrities.  Since then you learned that you could describe and compare growth by understanding a little bit about different types of growth, and using a growth model to make a prediction.

Now how about you be the one tracking the celebrities?  Imagine that you started a celebrity blog a few years back.  You write about hot news in Hollywood, and what your favorite celebs are up to.  Between 2010 and 2012, your blog followers grew at a rate of 5% to 9,500 people.  You hope to have 20,000 followers by 2020.  If this growth rate continues, will you meet your goal?

Because you know that the number of followers is growing at 5%, you know you’re dealing with exponential growth.  In order to develop a model, you need an initial value. To determine the initial value, you can use the number of followers in 2012 ($P_0$ = 9,500), and the rate of growth, which is 5% ($r$ = 5%), and solve for $P_0$.

 $P_n=\left(1+r\right)^nP_0$ Start with the explicit form of the exponential growth equation. $P_n=\left(1+0.05\right)^nP_0$ Substitute the growth rate, 5%, written as a decimal for $r$. $P_n=\left(1+0.05\right)^n9,500$ Substitute the initial value, which is the number of followers in 2012, for $P_0$. $P_8=\left(1+0.05\right)^89,500$ Substitute the number of years between 2012 and 2020 for $n$. $P_8=\left(1.05\right)^89,500\approx14,000$ Evaluate.

Unfortunately, you won’t quite meet your goal, but you’ll be close.

While online, you happen to notice that a different celebrity blogger is experiencing amazing success.  She appeared on the scene only nine months ago, but in that short time it grew exponentially from an initial following of 5,000 people to 12,000.  At what rate of growth is her following increasing?

 $P_0=5,000$ Determine the initial value. $P_9=\left(1+r\right)^9P_0$ Again use the explicit form of the exponential growth equation, but this time $t=9$ months. $12,000=\left(1+r\right)^95,000$ Substitute the initial value, which is the number of followers when she started, for $P_0$. ${\large\frac{12,000}{5,000}}=\left(1+r\right)^9$ Divide both sides by 5,000. $\sqrt[9]{12,000/5,000}=(1+r)^9$ Take the ninth root of both sides. $r=\sqrt[9]{12,000/5,000}-1\approx0.10$ Solve for $r$.

So her blog is growing at a rate of 10%.  What’s she got that you don’t have?  By comparing the rates of growth, you know it is definitely something.  Now you just have to figure out what it is and jazz up your blog to get followers breaking down the internet to read what you have to say!