In 2009,2012, and 2015, the number (in millions) of smart...

In $2009,2012,$ and $2015,$ the number (in millions) of smart phones sold in the world was $172.4,680.1,$ and $1423.9,$ respectively. (a) 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, $S(t)=\frac{M}{1+A e^{-k t}},$ that fits the given data points. (b) What is the long-term expected maximum number of smart phones sold annually? That is, what is $\lim _{t \rightarrow \infty} S(t) ?$ (c) In what year does the model predict that smart-phone sales will reach $98 \%$ of the expected maximum?

In $2009,2012,$ and $2015,$ the number (in millions) of smart phones sold in the world was $172.4,680.1,$ and $1423.9,$ respectively.
(a) 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, $S(t)=\frac{M}{1+A e^{-k t}},$ that fits the given data points.
(b) What is the long-term expected maximum number of smart phones sold annually? That is, what is $\lim _{t \rightarrow \infty} S(t) ?$
(c) In what year does the model predict that smart-phone sales will reach $98 \%$ of the expected maximum?

Key Concepts

Long-Term Behavior of the Logistic Function

The long-term behavior of a logistic function is characterized by its convergence to the carrying capacity M as time increases, because the exponential decay term approaches zero. This asymptotic approach indicates that once the system nears M, additional time results in negligible changes, reflecting market saturation or resource limitations.

Parameter Estimation from Data

Determining the parameters A and k in a logistic model involves using data points at different time instances to create equations that can be solved simultaneously. This estimation process ensures that the model accurately reflects the observed behavior and progression of the system, such as sales figures over specific years.

Exponential Decay Component

The term A·e^(–k·t) in the logistic model represents an exponentially decaying factor that modulates the initial growth phase. It controls how rapidly the growth slows down as the system approaches its carrying capacity, influencing both how the model starts (through A) and the rate of transition (through k).

Carrying Capacity (M)

In the logistic model, the parameter M represents the carrying capacity, which is the theoretical maximum value the function can reach as time goes to infinity. This parameter is crucial because it sets the upper limit for the modeled process, indicating, for example, the maximum number of smart phones that can be sold annually in the long run.

Logistic Growth Model

A logistic growth model describes processes where growth accelerates rapidly at first but then slows down as the system approaches a maximum limit. It is commonly represented by the formula S(t) = M / (1 + A·e^(–k·t)), combining exponential growth with a limiting factor, making it suitable for modeling saturated growth scenarios such as market penetration.

Please give Ace some feedback