Chapter 7.7, Problem 1MP

To determine

To check: The given statement: “The compound interest formula is an exponential growth function” is true or false.

Answer to Problem 1MP

True.

Explanation of Solution

Given Information: The compound interest formula is $A=P{\left(1+\frac{r}{n}\right)}^{nt}$

Formula used:

An exponential growth can be modeled by the function $y=a\cdot b^x$ , where $a>0$ and $\text{b>1}$ .

The base b is called growth factor.

The compound interest formula, $A=P{\left(1+\frac{r}{n}\right)}^{nt}$ is an exponential growth function since

  $b=1+\frac{r}{n}$ is clearly greater than one.

∴ The given statement: The compound interest formula is an exponential growth function is true.

(b)

To determine

To check the given statement: “The compound interest formula is an exponential decay function” is true or false

Answer to Problem 1MP

False.

Explanation of Solution

Given Information: The compound interest formula is $A=P{\left(1+\frac{r}{n}\right)}^{nt}$

Formula used:

An exponential decay can be modeled by the function $y=a\cdot b^x$ , where $a>0$ and $0<b<1$ .

The base b is called decay factor.

The compound interest formula, $A=P{\left(1+\frac{r}{n}\right)}^{nt}$ is not an exponential decay function since

  $b=1+\frac{r}{n}$ is not less than one.

∴ The given statement: The compound interest formula is an exponential decay function is false

(c)

To determine

To check the given statement: “For the CD from Bank West, $n=12$ ” is true or false

Answer to Problem 1MP

False.

Explanation of Solution

Given Information: Details of the West Bank is shown below:

Bank	CD Length	AnnualInterest Rate	Frequency of compounding
Bank West	6 years	3.8	Quarterly

In the formula for compound interest, $A=P{\left(1+\frac{r}{n}\right)}^{nt}$ , n represents number of times interest applied per time period.

For Bank West, it is given that the interest is compounded quarterly.

∴ The number of times interest applied per time period is 4

That is, $n=4$

∴The given statement: For the CD from Bank West, $n=12$ is false.

(d)

To determine

To check the given statement: “For the CD from First Bank, $nt=60$ ” is true or false

Answer to Problem 1MP

True.

Explanation of Solution

Given Information: Details of the First Bank is shown below:

Bank	CD Length	AnnualInterest Rate	Frequency of compounding
First Bank	5 years	4.3	Monthly

In the formula for compound interest,$A=P{\left(1+\frac{r}{n}\right)}^{nt}$ , n represents number of times interest applied per time period and t represents the time period.

For the First Bank it is given that the interest is compounded monthly and time period of the CD is 5 years.

That is, $n=5$ and $t=12$

∴ $nt=5\times 12=60$

∴ The given statement: For the CD from Bank First $nt=60$ is true.

(e)

To determine

To check the given statement: “The growth /decay factor for the CD from Bank West is ${\left(1+\frac{0.038}{4}\right)}^{24}$ ” is true or false

Answer to Problem 1MP

True.

Explanation of Solution

Given Information: Details of the West Bank is shown below:

Bank	CD Length	AnnualInterest Rate	Frequency of compounding
Bank West	6 years	3.8	Quarterly

The compound interest formula, $A=P{\left(1+\frac{r}{n}\right)}^{nt}$ is an exponential growth function [From part (a)]

n represents number of times interest applied per time period and t represents the time period.

For Bank West, it is given that the interest is compounded quarterly and time period is 6 years.

∴ The growth factor ${\left(1+\frac{r}{n}\right)}^{nt}$ = ${\left(1+\frac{3.8}{4\times 100}\right)}^{4\times 6}$

  = ${\left(1+\frac{0.038}{4}\right)}^{24}$

∴ The given statement: The growth /decay factor for the CD from Bank West is ${\left(1+\frac{0.038}{4}\right)}^{24}$ is true.

(f)

To determine

To check the given statement: “The growth /decay factor for the CD from Bank West is ${\left(1+\frac{0.038}{4}\right)}^4$ ” is true or false

Answer to Problem 1MP

False.

Explanation of Solution

From part (e) it has been find that the growth factor

for the CD from Bank West is ${\left(1+\frac{0.038}{4}\right)}^{24}$

∴ The given statement the growth /decay factor for the CD from Bank West is ${\left(1+\frac{0.038}{4}\right)}^4$ is false.

(g)

To determine

To check the given statement: “The growth /decay factor for the CD from Bank First is $0.0031\overline{6}$ ” is true or false

Answer to Problem 1MP

Explanation of Solution

Given Information: Details of the First Bank is shown below:

Bank	CD Length	AnnualInterest Rate	Frequency of compounding
First Bank	5 years	4.3	Monthly

For the First bank:

  $\begin{array}{l}n=12\\ r=4.3\\ t=5\end{array}$

Growth Factor for the CD from First Bank = ${\left(1+\frac{r}{n}\right)}^{nt}$

   = ${\left(1+\frac{5}{12}\right)}^{12\times 5}$

  = ${\left(1+0.416\right)}^{60}$ ( greater than 1)

Clearly the Growth Factor for the CD from First Bank is not equal to $0.0031\overline{6}$

∴The given statement: “The growth /decay factor for the CD from Bank First is $0.0031\overline{6}$ ” is false