Chapter 3.5, Problem 18E

To determine

Complete the given table for time $t$ .

Answer to Problem 18E

The model is $t=1.099r^{-1}$ .

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.

The complete table is

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

The graph for the data points is

  

By the use of regression , the model fits the data is

  $t=1.099r^{-1}$ .

Hence the model is $t=1.099r^{-1}$ .