An internet analytics company measured the number of people watching a video posted on a social media platform. The company found 125 people had watched the video and that the number of people who had watched it was increasing by 30% every 3 hours.
(a)
Write an exponential function for the number of people A who had watched the video n hours after the initial observation.

We can model the growth of the number of people watching the video using an exponential function of the form:

\[
A(n) = A_0 \cdot (1 + r)^n
\]

where:
- \(A(n)\) is the number of people who have watched the video after \(n\) hours.
- \(A_0\) is the initial number of people who have watched the video at the start (at \(n = 0\)).
- \(r\) is the rate of growth per time period (in this case, 30% per 3 hours, so \(r = 0.30\)).
- \(n\) is the number of 3-hour periods.

First, note that the growth rate of 30% happens every 3 hours, so we need to adjust the function to reflect that growth occurs over multiples of 3 hours. For every 3 hours, we multiply the initial number of people by \(1.30\). 

However, if we want the function to reflect the growth for any number of hours \(n\), we convert the time from 3-hour intervals to a function of hours. Each 3-hour interval is \(\frac{n}{3}\), so the function becomes:

\[
A(n) = 125 \cdot (1.30)^{\frac{n}{3}}
\]

Thus, the exponential function for the number of people who have watched the video \(n\) hours after the initial observation is:

\[
A(n) = 125 \cdot (1.30)^{\frac{n}{3}}
\]

Would you like further clarification or have any questions?

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Here are five questions that expand on this information:
1. How many people would have watched the video after 6 hours?
2. How long would it take for 500 people to watch the video?
3. How does changing the rate of increase affect the function?
4. What would the model look like if the video lost viewers instead of gaining them?
5. How would you model growth if the rate were 25% every 4 hours?

**Tip:** When working with exponential growth, always check whether the time interval matches the growth rate's time frame for accurate modeling!

An internet analytics company measured the number of people watching a video posted on a social media platform. The company found 125 people had watched the video and that the number of people who had watched it was increasing by 30% every 3 hours.
(a) Write an exponential function for the number of people A who had watched the video n hours after the initial observation.

Learn how to model the exponential growth of a social media video’s view count, starting with 125 views and increasing by 30% every 3 hours. This problem covers key algebraic concepts, including exponential growth functions. Suitable for students in Grades 9-12.

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