Chapter 7.4, Problem 46E

a.

To determine

To prove: That it takes $t=\frac{\ln 2}{r}$ years for money to double.

Explanation of Solution

Given information: That it takes $t=\frac{\ln 2}{r}$ years for money to double if it is invested at annual

Interest rate r compounded continuously.

Formula used: The amount of money in the account after t year is then

  $A(t)=A_0{e}^{rt}$ .

Interest paid according to this formula is said to be compounded continuously. The number r is the continuous Interest rate.

Proof:

The amount of money in the account after t year is then

  $A(t)=A_0{e}^{rt}$

  $\begin{array}{l}A(t)=A_0{e}^{rt}\\ A(t)=2A_0\\ 2A_0=A_0{e}^{rt}\\ e^{rt}=2\\ rt=\ln 2\\ t=\frac{\ln 2}{r}\end{array}$

b.

To determine

To graph: Graph the functions

  $y_1=\frac{\ln 2}{r},y_2=\frac{70}{100r},y_3=\frac{72}{100r}$

Explanation of Solution

Given information: Graph the functions

  $y_1=\frac{\ln 2}{r},y_2=\frac{70}{100r},y_3=\frac{72}{100r}$ in the $\left[0,0.1\right]$ by $\left[0,100\right]$

Graph:

functions

  $y_1=\frac{\ln 2}{r},y_2=\frac{70}{100r},y_3=\frac{72}{100r}$

  

c.

To determine

To calculate: Explain why these two rules of thumb for mental computation are reasonable.

Explanation of Solution

Given information: Use the rules to estimate how long it takes to double money at $5\%$ compounded continuously.

Formula used: The amount of money in the account after t year is then

  $A(t)=A_0{e}^{rt}$

Interest paid according to this formula is said to be compounded continuously. The number r is the continuous Interest rate.

Calculation: the functions

  $y_1=\frac{\ln 2}{r},y_2=\frac{70}{100r},y_3=\frac{72}{100r}$

  $\ln 2$ is approximately $0.7$ or $\frac{70}{100}$ and $72$ is not much larger and makes the math easier to compute mentally

d.

To determine

To calculate: Use the rules to estimate how long it takes to double money.

Answer to Problem 46E

Money will double in $t=14.4\text{ Years}$ .

Explanation of Solution

Given information: Use the rules to estimate how long it takes to double money at $5\%$ compounded continuously.

Formula used: The amount of money in the account after t year is then

  $A(t)=A_0{e}^{rt}$ .

Interest paid according to this formula is said to be compounded continuously. The number r is the continuous Interest rate.

Calculation: Use the rules to estimate how long it takes to double money at $5\%$ compounded continuously

The amount of money in the account after t year is then

  $A(t)=A_0{e}^{rt}$

  $\begin{array}{l}A(t)=A_0{e}^{rt}\\ A(t)=2A_0\\ 2A_0=A_0{e}^{rt}\\ e^{rt}=2\\ rt=\ln 2\\ t=\frac{\ln 2}{r}\\ r=5\%\\ t=\frac{\ln 2}{0.05}\\ t=14.4\text{ Years}\end{array}$

e.

To determine

To calculate: Use the rules to estimate how long it takes to triple money.

Answer to Problem 46E

Money will triple in $t=\frac{\ln 3}{r}$ .

Explanation of Solution

Given information: Use the rules to estimate how long it takes to triple money.

Formula used: The amount of money in the account after t year is then

  $A(t)=A_0{e}^{rt}$

Interest paid according to this formula is said to be compounded continuously. The number r is the continuous Interest rate.

Calculation: Use the rules to estimate how long it takes to double money at $5\%$ compounded continuously.

The amount of money in the account after t year is then

  $A(t)=A_0{e}^{rt}$

  $\begin{array}{l}A(t)=A_0{e}^{rt}\\ A(t)=2A_0\\ 3A_0=A_0{e}^{rt}\\ e^{rt}=3\\ rt=\ln 3\\ t=\frac{\ln 3}{r}\end{array}$