Chapter 6.4, Problem 46E

(a)

To determine

The time for money to get double if invested at rate r compounded continuously.

Answer to Problem 46E

The time required to double the money is $\frac{\ln 2}{r}$

Explanation of Solution

Given:

The money invested compounded annually.

Calculation:

The formula to calculate compound interest is:

  $A=P{e}^{rt}$

Where:

A is the amount

P is the principal amount.

R is the rate.

T is the time for which the money is invested.

Hence, the time required for money to get double is:

  $\begin{array}{l}A=P{e}^{rt}\\ 2P=P{e}^{rt}\\ 2=e^{rt}\\ \ln 2=rt\\ t=\frac{\ln 2}{r}\end{array}$

Conclusion:

The time required to double the money is $\frac{\ln 2}{r}$

(b)

To determine

Graph the functions.

Answer to Problem 46E

The given functions are graphed.

Explanation of Solution

Given: $y_1=\frac{\ln 2}{r},y_2=\frac{70}{100r},\text{and}{y}_3=\frac{72}{100r}$

Calculation:

  

Conclusion:

The given functions are graphed.

(c)

To determine

The reason for use of two rules of thumb for computation.

Answer to Problem 46E

The rule gives the best approximate answer.

Explanation of Solution

Given:

  $y_1=\frac{\ln 2}{r},y_2=\frac{70}{100r},\text{and}{y}_3=\frac{72}{100r}$

Calculation:

  $\begin{array}{l}t=\frac{\ln 2}{r}\\ \ln 2=0.6931\\ \frac{70}{100}=0.7\\ \frac{72}{100}=0.72\end{array}$

The values obtained are very close to each other and the answer is better. Therefore, this rule is best.

Conclusion:

The rule gives the best approximate answer.

(d)

To determine

The time at which the money gets double at $5\%$ compounded annually.

Answer to Problem 46E

The time at which the money gets double at $5\%$ compounded annually is approximately $14\text{ yrs}$ .

Explanation of Solution

Given: $f(t)=\frac{4}{1+t^2}$ and $G(x)={\displaystyle \underset{0}{\overset{x}{\int }}f(t)dt}$

Calculation:

The actual time takes:

  $\text{t = }\frac{\ln 2}{0.5}=13.86$

The approximate time takes:

  $\begin{array}{l}t=\frac{70}{100(0.05)}=14\text{ yrs}\\ \text{t = }\frac{72}{5}=14.4\text{ yrs}\end{array}$

(e)

To determine

The rule needs to be find in order to calculate the time required to triple the money.

Answer to Problem 46E

The time at which the money gets triple is $\frac{\ln 3}{r}$

Explanation of Solution

Given: $f(t)=\frac{4}{1+t^2}$ and $G(x)={\displaystyle \underset{0}{\overset{x}{\int }}f(t)dt}$

Calculation:

  $\begin{array}{l}A=P{e}^{rt}\\ 3P=P{e}^{rt}\\ \ln 3=rt\\ t=\frac{\ln 3}{r}\end{array}$