in 2009, 2012, 2015, the number (in millions) of smart phones sold in the world was 167.4, 675.1, and 1418.9, 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 logistical 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. 

To find the parameters \( M \), \( A \), and \( k \) in the logistic model:

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

we will use the given data points:

- In 2009: \( S(0) = 167.4 \) million
- In 2012: \( S(3) = 675.1 \) million
- In 2015: \( S(6) = 1418.9 \) million

### Step 1: Determine \( M \)
The parameter \( M \) represents the carrying capacity, which is the maximum number of smartphones that can be sold as \( t \to \infty \). Observing the data, the number of smartphones sold increases over time, but it should eventually level off. Based on the growth pattern, we estimate \( M \) to be slightly above the highest value provided, \( S(6) = 1418.9 \) million. We will approximate \( M \) based on the logistic growth.

### Step 2: Formulate Equations
Using the logistic model formula, we can substitute the known data points to set up the system of equations.

1. For \( t = 0 \) (2009):
\[
167.4 = \frac{M}{1 + A e^{-k(0)}} = \frac{M}{1 + A}
\]

2. For \( t = 3 \) (2012):
\[
675.1 = \frac{M}{1 + A e^{-3k}}
\]

3. For \( t = 6 \) (2015):
\[
1418.9 = \frac{M}{1 + A e^{-6k}}
\]

### Step 3: Solve the System of Equations
We'll solve this system of nonlinear equations using numerical methods, as there isn't a simple algebraic solution.

Let's proceed with the calculations to find approximate values for \( M \), \( A \), and \( k \).The approximate values for the parameters in the logistic model \( S(t) = \frac{M}{1 + A e^{-kt}} \) that fits the given data points are:

- \( M \approx 1843.3 \) (in millions, representing the carrying capacity)
- \( A \approx 10.011 \)
- \( k \approx 0.585 \)

These parameters provide a logistic growth model that approximates the sales of smartphones over time, based on the data from 2009, 2012, and 2015.

Would you like to see the detailed steps for solving this system, or do you have any other questions?

### Related Questions:
1. How does the logistic growth model behave as \( t \to \infty \)?
2. What would the estimated number of smartphones sold be in 2020 using this model?
3. How sensitive is the model to changes in \( A \) and \( k \)?
4. Can the logistic model be used to predict future sales accurately beyond 2015?
5. What assumptions does the logistic model make about the market saturation of smartphones?

### Tip:
In logistic growth, the value of \( k \) controls the growth rate. A higher \( k \) means quicker saturation, while a lower \( k \) indicates slower growth towards the carrying capacity \( M \).

In 2009, 2012, and 2015, the number (in millions) of smartphones sold in the world was 167.4, 675.1, and 1418.9, 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 logistical model, S(t) = M / (1 + A * e^-kt), that fits the given data points. Approximate A and k to three decimal places and M to one decimal place.

Discover how to determine the parameters M, A, and k in a logistic growth model that fits smartphone sales data from 2009, 2012, and 2015. Explore the mathematical concepts of logistic growth and exponential decay, including detailed calculations and approximations. Ideal for advanced high school or college students.

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