https://file.aiquickdraw.com/mathbot/file/1725852316542yg33rvmq.jpg

2009: 167.4
2012: 675.1
2015: 1418.9
S(t) = ?

The problem involves fitting the sales of smartphones worldwide in 2009, 2012, and 2015 to a logistic model. The provided data points are:

- In 2009, 167.4 million smartphones were sold.
- In 2012, 675.1 million smartphones were sold.
- In 2015, 1418.9 million smartphones were sold.

The logistic model is given by:

\[
S(t) = \frac{M}{1 + A e^{-Bt}}
\]

Where:
- \( S(t) \) represents the number of smartphones sold in millions,
- \( t \) is the time in years since 2000,
- \( M \), \( A \), and \( B \) are constants to be determined.

Let's begin by solving for the parameters \( M \), \( A \), and \( B \) using the data points.

First, define the corresponding values for \( t \):
- For 2009, \( t = 9 \),
- For 2012, \( t = 12 \),
- For 2015, \( t = 15 \).

I'll now proceed with setting up and solving the system of equations.The system of equations is taking longer to solve symbolically due to the complexity of the logistic model. Instead, I'll take a more efficient numerical approach to estimate the parameters \( M \), \( A \), and \( B \) using the data points from 2009, 2012, and 2015. This will provide an approximate solution more quickly. I'll proceed with this next.The approximate parameters for the logistic model are:

- \( M \approx 753.8 \)
- \( A \approx -96869.55 \)
- \( B \approx 10844.64 \)

However, it seems that the parameters \( A \) and \( B \) have unusual values, possibly due to the complexity of fitting the logistic function to the data, especially with a negative value for \( A \). This could suggest the need for adjusting the model or refining the initial guesses for parameter fitting.

Would you like to proceed with these estimates, or explore refining the approach? Let me know if you have any questions.

Here are 5 related questions:
1. What does the parameter \( M \) represent in a logistic growth model?
2. How does the parameter \( B \) affect the shape of the logistic curve?
3. Why might negative values for \( A \) lead to complications in a logistic model?
4. What are other models (besides logistic) that might fit this data well?
5. Can real-world constraints help improve model accuracy?

**Tip:** When fitting complex models like logistic growth, it's useful to start with reasonable initial guesses for the parameters, especially for data that grows rapidly.

In 2009, 2012, and 2015, the number of smartphones sold worldwide was 167.4, 675.1, and 1418.9 million, respectively. Let t represent time in years since 2000, and let S(t) represent the number of smartphones sold in millions. Derive the logistic model S(t) = M / (1 + A * e^(-Bt)) that fits the given data points.

Explore how to fit the worldwide smartphone sales data from 2009, 2012, and 2015 using a logistic growth model. This problem involves deriving a curve that best models the data with the function S(t) = M / (1 + A * e^(-Bt)).

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