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 $M$.