Chapter 7.8, Problem 2GP

To determine

To write: A function to represent the number of subscribers of app $A$ after $t$ years and then write an equivalent function that represents the subscribers to app $A$ with only one compounding per year, determine the effective yearly rate of growth of the subscribers to app $A$ , and at last determine the faster growing app.

Answer to Problem 2GP

  $A(t)=P{(1.015)}^{12t}$

  $A(t)=P{(1.19561817)}^t$

Number of subscribers to app $A$ is increasing at a faster rate.

Explanation of Solution

Given information: Subscribers to app $A$ increase by $1.5\%$ each month, subscribers to app $B$ is growing by $16\%$ per year.

Calculation:

  $A(t)=P{(1+\frac{r}{n})}^{nt}(Formulaofcompound\mathrm{int}erest)$

Substituting the values in the formula to derive the required function,

  $\begin{array}{}$ A(t)=P{(1+\frac{0.015\times 12}{12})}^{12t}$$ A(t)=P{(1.015)}^{12t}$\end{array}$

Writing an equivalent function that represents the subscribers to app $A$ with only one compounding per year,

  $\begin{array}{}$ A(t)=P{({(1.015)}^{12})}^t$$ A(t)=P{(1.19561817)}^t$\end{array}$

Calculating effective yearly rate of growth of subscribers to app $A$ ,

  $\begin{array}{l}1+r=1.19561817\\ r=0.1956\\ r=19.56\%\end{array}$

Effective yearly rate of growth of subscribers to app $A$ is $19.56\%$ .

Yearly rate of growth of subscribers to app $B$ is $16\%$ , thus, number of subscribers to app $A$ is increasing at a faster rate.