In 2009, 2012, and 2015, the number (in millions) of smart phones sold in the world was 165.4, 673.1, and 1416.9, respectively. Let t represent time in years since 2009, and let S represent the number of smart phones sold in millions. Determine M, A, and k for a logistic model, S(t) = M/1+Ae^-kt, that fits the given data points. Approximate A and k to three decimal place and M to one decimal place

To model the data using the logistic function \( S(t) = \frac{M}{1 + Ae^{-kt}} \), where:

- \( S(t) \) is the number of smartphones sold (in millions) at time \( t \) years since 2009,
- \( M \) is the carrying capacity (maximum number of smartphones that can be sold),
- \( A \) and \( k \) are parameters to be determined.

We are given the following data points:

1. In 2009 (when \( t = 0 \)): \( S(0) = 165.4 \) million,
2. In 2012 (when \( t = 3 \)): \( S(3) = 673.1 \) million,
3. In 2015 (when \( t = 6 \)): \( S(6) = 1416.9 \) million.

### Step 1: Logistic Model Setup
The general form of the logistic equation is:

\[
S(t) = \frac{M}{1 + Ae^{-kt}}
\]

At \( t = 0 \), this simplifies to:

\[
S(0) = \frac{M}{1 + A}
\]

So, we know that \( S(0) = 165.4 \), which gives:

\[
165.4 = \frac{M}{1 + A}
\]

From this, we can solve for \( A \) in terms of \( M \):

\[
1 + A = \frac{M}{165.4}, \quad A = \frac{M}{165.4} - 1 \tag{1}
\]

### Step 2: Using the Other Data Points
Now, for \( t = 3 \) (in 2012) and \( t = 6 \) (in 2015), we substitute the given values into the logistic equation.

For \( t = 3 \):

\[
673.1 = \frac{M}{1 + A e^{-3k}} \tag{2}
\]

For \( t = 6 \):

\[
1416.9 = \frac{M}{1 + A e^{-6k}} \tag{3}
\]

### Step 3: Solving the System of Equations
We now have three equations: one from \( t = 0 \), one from \( t = 3 \), and one from \( t = 6 \). These can be solved simultaneously to find the values of \( M \), \( A \), and \( k \). I'll calculate them step by step.

Let's solve this.The parameters for the logistic model \( S(t) = \frac{M}{1 + Ae^{-kt}} \) that fits the given data are:

- \( M = 1835.3 \) (the carrying capacity in millions),
- \( A = 10.096 \),
- \( k = 0.589 \).

Would you like more details on this process or have any specific questions?

Here are 5 related questions to deepen your understanding:
1. How is the logistic growth model used in other real-world applications?
2. What happens to the number of smartphones sold as \( t \) increases indefinitely?
3. Can the logistic model be applied to predict smartphone sales beyond 2015?
4. How does changing the value of \( k \) affect the growth curve of the logistic function?
5. What is the significance of the parameter \( A \) in shaping the logistic curve?

**Tip:** The logistic model is often used when growth initially accelerates but slows down as it approaches a maximum value.

In 2009, 2012, and 2015, the number (in millions) of smartphones sold in the world was 165.4, 673.1, and 1416.9, respectively. Let t represent time in years since 2009, and let S represent the number of smartphones sold in millions. Determine M, A, and k for a logistic model, S(t) = M/(1+Ae^-kt), that fits the given data points. Approximate A and k to three decimal places and M to one decimal place.

Learn how to determine the parameters M, A, and k for a logistic growth model S(t) = M/(1+Ae^-kt) based on smartphone sales data from 2009 to 2015. The solution includes step-by-step calculations and approximates A and k to three decimal places and M to one decimal place. Suitable for Grades 10-12 and college-level math students.

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