The number of people using a social media application is declining by 15% monthly.  Use the model A(t)=Ie^rt where r is the growth rate/decay. If initially there are 22,000,000 users, find the number of users after one year 

To find the number of users after one year using the given model \( A(t) = Ie^{rt} \), we need to follow these steps:

1. Identify the initial number of users (\(I\)).
2. Determine the decay rate (\(r\)).
3. Calculate the number of users after one year (\(t = 1\) year).

Given:
- Initial number of users, \(I = 22,000,000\)
- Monthly decline rate is 15%. This is a decay, so \(r\) will be negative.

First, convert the monthly decay rate to an annual decay rate. Since the decline is 15% per month, we need to find the annual decay rate.

The monthly decay rate can be written as:
\[ 1 - 0.15 = 0.85 \]

The annual decay rate is:
\[ (0.85)^{12} \]

We need to convert this annual decay factor to an exponential decay rate \(r\):
\[ 0.85^{12} = e^{r \cdot 12} \]
Taking the natural logarithm on both sides:
\[ \ln(0.85^{12}) = r \cdot 12 \]
\[ \ln(0.85^{12}) = 12 \ln(0.85) \]
\[ r = \frac{12 \ln(0.85)}{12} \]
\[ r = \ln(0.85) \approx -0.1625 \]

Now, using \(r = -0.1625\), \(I = 22,000,000\), and \(t = 1\) year, we can calculate the number of users after one year:

\[ A(1) = 22,000,000 \cdot e^{-0.1625 \cdot 1} \]

\[ A(1) = 22,000,000 \cdot e^{-0.1625} \]

Using a calculator to find \( e^{-0.1625} \approx 0.8504 \):

\[ A(1) = 22,000,000 \cdot 0.8504 \approx 18,708,800 \]

So, the number of users after one year is approximately 18,708,800.

Would you like more details or have any questions?

Here are some additional questions you might ask:

1. How do we convert a monthly decay rate to an annual decay rate?
2. Can you explain the natural logarithm function in this context?
3. How do you interpret the decay factor in exponential decay?
4. What are some real-life applications of exponential decay?
5. How would the calculation change if the monthly decline rate was different?

**Tip:** Always double-check your calculations, especially when working with exponents and logarithms, to ensure accuracy.

Learn how to calculate the number of users after one year using the exponential decay model A(t) = I * e^(rt). Initial users: 22,000,000. Monthly decline rate: 15%. Follow step-by-step calculations to find the approximate number of users after one year. Suitable for advanced high school students and beyond.

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