Chapter 3.5, Problem 17E

a.

To determine

Complete the given table for time $t$ .

Answer to Problem 17E

The complete table is given below.

$r$	$t=\frac{\ln 3}{r}$
$\begin{array}{l}2\%\\ 4\%\\ 6\%\\ 8\%\\ 10\%\\ 12\%\end{array}$	$\begin{array}{l}54.93\\ 27.465\\ 18.31\\ 13.7327\\ 10.986\\ 9.155\end{array}$

Explanation of Solution

Given:

  $p$ dollars to triple when interest is compounded continuously.

The given table is

  

Calculation:

  $p$ dollars to triple when interest is compounded continuously.

  $A=3p$

Use formula.

  $\begin{array}{l}A=p{e}^{rt}\\ \Rightarrow 3p=p{e}^{rt}\\ \Rightarrow \frac{3p}{p}=\frac{p}{p}{e}^{rt}\\ \Rightarrow 3=e^{rt}\end{array}$

Taking natural log on both sides.

  $\Rightarrow \ln 3=\ln e^{rt}$

Use power property.

  $\begin{array}{l}\Rightarrow \ln 3=rt\\ \Rightarrow \frac{\ln 3}{r}=\frac{rt}{r}\\ \Rightarrow \frac{\ln 3}{r}=t\\ \Rightarrow t=\frac{\ln 3}{r}\end{array}$

Substitute all the $r$ values in the relation $t=\frac{\ln 3}{r}$ and find all the corresponding t values.

Hence the complete table is given below.

$r$	$t=\frac{\ln 3}{r}$
$\begin{array}{l}2\%\\ 4\%\\ 6\%\\ 8\%\\ 10\%\\ 12\%\end{array}$	$\begin{array}{l}54.93\\ 27.465\\ 18.31\\ 13.7327\\ 10.986\\ 9.155\end{array}$

b.

To determine

Complete the given table for time $t$ .

Answer to Problem 17E

The complete table is given below.

$r$	$t=\frac{\log 3}{\log \left(1+r\right)}$
$\begin{array}{l}2\%\\ 4\%\\ 6\%\\ 8\%\\ 10\%\\ 12\%\end{array}$	$\begin{array}{l}55.478\\ 28.011\\ 18.85\\ 14.27\\ 11.5267\\ 9.69\end{array}$

Explanation of Solution

Given:

  $p$ dollars to triple when interest is compounded annually.

The given table is

  

Calculation:

  $p$ dollars to triple when interest is compounded annually.

  $A=3p$

Use formula

  $\begin{array}{l}A=p{\left(1+\frac{r}{n}\right)}^{nt}\\ 3p=p{\left(1+\frac{r}{1}\right)}^{1t}\\ \Rightarrow 3p=p{\left(1+r\right)}^t\\ \Rightarrow 3\frac{p}{p}=\frac{p}{p}{\left(1+r\right)}^t\\ \Rightarrow 3={\left(1+r\right)}^t\end{array}$

Taking log on both sides.

  $\Rightarrow \log 3=\log {\left(1+r\right)}^t$

Use power rule.

  $\begin{array}{l}\Rightarrow \log 3=t\log \left(1+r\right)\\ \Rightarrow \frac{\log 3}{\log \left(1+r\right)}=t\frac{\log \left(1+r\right)}{\log \left(1+r\right)}\\ \Rightarrow t=\frac{\log 3}{\log \left(1+r\right)}\end{array}$

Substitute all the $r$ values in the relation $t=\frac{\log 3}{\log \left(1+r\right)}$ and find all the corresponding t values.

Hence the complete table is given below.

$r$	$t=\frac{\log 3}{\log \left(1+r\right)}$
$\begin{array}{l}2\%\\ 4\%\\ 6\%\\ 8\%\\ 10\%\\ 12\%\end{array}$	$\begin{array}{l}55.478\\ 28.011\\ 18.85\\ 14.27\\ 11.5267\\ 9.69\end{array}$