The percentage of people age 12 years and older using social...

The percentage of people age 12 years and older using social network sites and/or services "several" times a day from 2009 through 2013 grew at the rate of $$R(t)=5.92 t^{-0.168} \quad(1 \leq t \leq 5)$$ percent/year in year $t,$ with $t=1$ corresponding to $2009 .$ The percentage of people age 12 years and older using social network sites and/or services in 2009 was $7 \%$. a. Find an expression giving the percentage of people age 12 years and older using social network sites and/or services in year $t(1 \leq t \leq 5)$ b. According to this model, what percentage of people age 12 years and older used social network sites and/or services in $2013 ?$

The percentage of people age 12 years and older using social network sites and/or services "several" times a day from 2009 through 2013 grew at the rate of
$$R(t)=5.92 t^{-0.168} \quad(1 \leq t \leq 5)$$
percent/year in year $t,$ with $t=1$ corresponding to $2009 .$ The percentage of people age 12 years and older using social network sites and/or services in 2009 was $7 \%$.
a. Find an expression giving the percentage of people age 12 years and older using social network sites and/or services in year $t(1 \leq t \leq 5)$
b. According to this model, what percentage of people age 12 years and older used social network sites and/or services in $2013 ?$

Key Concepts

Differential Equations in Growth Models

Differential equations are used to model dynamic processes where a rate of change is described in relation to the current state of the system. Here, the differential equation relating the growth rate and the accumulated percentage allows us to predict the evolution of social network use over time.

Rate of Change

This concept refers to the function that measures how a quantity changes over time. In the context of the model, R(t) describes the instantaneous percentage increase per year in the use of social networks, serving as the driving force behind any subsequent growth equation.

Integration as an Accumulation Process

Integration is the mathematical process of summing up small increments of change over a period to obtain a total change. When given a rate function such as R(t), integration helps to determine the cumulative effect of that rate from a starting time to any later time.

Initial Value Problem

An initial value problem involves finding a function that satisfies a differential equation along with an initial condition. In this case, knowing that 7% of the population used social network sites in the initial year (t = 1) enables the determination of the constant of integration when solving for the percentage function.

Power Function Rate Models

The rate function R(t) is given as a power function of time, which means that the rate of increase decreases following a specific power law. This reflects non-linear growth characteristics, unlike constant exponential growth, and is crucial for accurately modeling systems where the growth rate slows over time.

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