Chapter 0, Problem 66RE

(a)

To determine

To find: $t$ as a function of $r$ .

Answer to Problem 66RE

  $t=\frac{\ln 2}{r}$

Explanation of Solution

Given information: An annual interest rate $R\text{\%}$ with double an investment in approximately $70/R$ years.

Calculation:

First, divide both sides by $P$ :

  $e^{rt}=2$

On both sides of the equation, take $\ln$ now.

  $\ln e^{rt}=\ln 2$

To use the logarithmic formulas, get at:

  $rt=\ln 2$

At last:

  $t=\frac{\ln 2}{r}$

Therefore, the required $t$ as a function of $r$ is $t=\frac{\ln 2}{r}$ .

(b)

To determine

To find: $r=R\text{\%}$ , write $t$ as a function of $R$.

Answer to Problem 66RE

  $t=\frac{69}{R}$

Explanation of Solution

Given information: An annual interest rate $R\text{\%}$ with double an investment in approximately $70/R$ years.

Calculation:

Given that dividing by $100$ is a percentage,

  $r=\frac{R}{100}$

From part a)

  $t=\frac{\ln 2}{r}$

Then substitute $r=\frac{R}{100}$ , get:

  $\begin{array}{c}t=\frac{\ln }{\frac{R}{100}}\\ \Rightarrow t=\frac{100\ln 2}{R}\\ \Rightarrow t=\frac{69}{R}\end{array}$

Therefore, the required $t$ as a function of $R$is $t=\frac{69}{R}$ .

(c)

To determine

To find: write $t$ as a function of $R$.

Answer to Problem 66RE

  $t=\frac{70}{R}$

Explanation of Solution

Given information: An annual interest rate $R\text{\%}$ with double an investment in approximately $70/R$ years.

Calculation:

From part a)

  $t=\frac{69}{R}$

Addition of $1$ results in:

  $t=\frac{70}{R}$

Therefore, the required $t$ as a function of $R$is $t=\frac{70}{R}$ .