Chapter 5.4, Problem 95E

To determine

The highest value of investment from $\left(a\right),\left(b\right)$ and $\left(c\right)$ where, $\left(a\right)$ double the invested amount, $\left(b\right)$ double the interest rate, $\left(c\right)$ double the number of years on investing $P$ dollars at an annual interest rate of $r$ , compounded continuously, for the time period of $t$ years.

For $e^{rt}>2$ , doubling either interest or the number of years would yield more money and for $e^{rt}<2$ , doubling the investment would yield more money.

Explanation:

Consider the invested amount $P$ dollars at an annual interest rate of $r$ , compounded continuously for the time period of $t$ years.

Now, the formula for continuous compounding is:

$A=P{e}^{rt}$ …… $\left(1\right)$

Where $A$ is the final amount, $P$ is principal, $r$ is rate of interest and $t$ is the time in years.

Now,

$\left(a\right)$ On doubling the invested amount,

Now, replace $P$ by $2P$ in the equation $\left(1\right)$ as:

$A=\left(2P\right)e^{rt}$

$A=2P{e}^{rt}$ …… $\left(2\right)$

Now,

$\left(b\right)$ On doubling the interest rate,

Now, replace r by 2r in the equation $\left(1\right)$ as:

$\begin{array}{c}A=P{e}^{2rt}\\ A=P{e}^{rt}{e}^{rt}\end{array}$

$A=e^{rt}\left(P{e}^{rt}\right)$ …… $\left(3\right)$

Now,

$\left(c\right)$ On doubling the number of years,

Now, replace $t$ by $2t$ in the equation $\left(1\right)$ as:

$\begin{array}{c}A=P{e}^{r2t}\\ A=P{e}^{2rt}\\ A=P{e}^{rt}{e}^{rt}\end{array}$

$A=e^{rt}\left(P{e}^{rt}\right)$ …… $\left(4\right)$

Now, take natural log on both sides of the equation $\left(2\right)$ as,

$\begin{array}{c}\ln A=\ln 2P{e}^{rt}\\ \ln A=\ln 2+\ln P+\ln e^{rt}\end{array}$

According to the inverse property of logarithmic functions:

${\log }_a{a}^x=x$ .

Hence,

$\ln e^{rt}=rt$

$\ln A=\ln 2+\ln P+rt$ …… $\left(5\right)$

Now, take natural log on both sides of the equation $\left(3\right)$ as,

$\begin{array}{l}\ln A=\ln \left[e^{rt}\left(P{e}^{rt}\right)\right]\\ \ln A=\ln e^{rt}+\ln \left(P{e}^{rt}\right)\end{array}$

$\ln A=\ln e^{rt}+\ln P+\ln e^{rt}$

According to inverse property of logarithmic functions:

${\log }_a{a}^x=x$ .

Hence,

$\ln A=rt+\ln P+rt$

$\ln A=2rt+\ln P$ …… $\left(6\right)$

Now, take natural log on both sides of the equation $\left(4\right)$ as,

$\begin{array}{l}\ln A=ln\left[e^{rt}\left(P{e}^{rt}\right)\right]\\ \ln A=\ln e^{rt}+\ln \left(P{e}^{rt}\right)\\ \ln A=\ln e^{rt}+\ln P+\ln e^{rt}\end{array}$

According to the inverse property of logarithmic functions:

${\log }_a{a}^x=x$ .

Hence,

$\ln A=rt+\ln P+rt$

$\ln A=2rt+\ln P$ …… $\left(7\right)$

Now, if $rt>\ln 2$

Then, clearly from equation $\left(5\right)$ the amount on doubling the investment yields more money.

Now, if $rt<\ln 2$

Then, clearly from equation $\left(6\right)\text{and}\left(7\right)$ the amount on doubling either interest or the number of years yields more money.