Chapter 6.4, Problem 44E

(a)

To determine

To explain : The Jhon Nepier’s answer.

Answer to Problem 44E

More specifically,  money will increase by a factor of roughly $2.718$ per year if  leave it alone.

Explanation of Solution

Given information:

An amount of money at $100\text{\%}$ yearly interest.

Calculation:

To use the formula:

  $P(t)=P_0{e}^{rt}$

  $P(t)$ is money after time $t$ and $r$ is the interest.

Put, $r=1$ :

  $P(t)=P_0{e}^t$

 See from the calculation that money is more sensitive right now than it was when interest rates were lower. This implies that money will increase over the same period of time.

More specifically,  money will increase by a factor of roughly $2.718$ per year if  leave it alone.

(b)

To determine

To find: The times it take to triple the money.

Answer to Problem 44E

The answer is: $1.0986\text{ years}$

Explanation of Solution

Given information:

An amount of money at $100\text{\%}$ yearly interest.

Calculation:

To use the formula:

  $A(t)=P{e}^{rt}$

Years $t$ and $r$ is the interest.

Here,

  $\begin{array}{c}A(t)=3P\\ r=100\text{\%}\\ =1.\end{array}$

  $\begin{array}{c}A(t)=P{e}^{rt}\\ \Rightarrow 3P=P{e}^{1\cdot t}\\ \Rightarrow 3P=P{e}^{1\cdot t}\\ \Rightarrow \ln 3=\ln e^t\\ \Rightarrow \ln 3=t\\ \Rightarrow t\approx 1.0986\text{ years}\end{array}$

Therefore the required $1.0986\text{ years}$ it take to triple the money.

(c)

To determine

To find: The earn in a year.

Answer to Problem 44E

Earn about $172\text{\%}$ of the initial investment.

Explanation of Solution

Given information:

An amount of money at $100\text{\%}$ yearly interest.

Calculation:

From part a) money will grow by a factor of $e\approx 2.72$ each year that means it increases by $272\text{\%}-100\text{\%}=172\text{\%}$ each year.

Earn about $172\text{\%}$ of the initial investment.