Chapter 2.7, Problem 64PS

Solution

(a.)

To Complete: The given table.

It has been determined that the value of the second deposit for year 3 is $1000g$ , its value for year 4 is $1000g^2$ and the value of the third deposit for year 4 is $1000g$ . These values complete the given table.

Given:

An amount of $\$1000$ is saved and deposited into a bank account at the end of each summer, each year towards buying a used car in four years.

The growth factor $1+r$ , where $r$ is the annual interest rate expressed as a decimal, is denoted by $g$ .

The following table shows the value of the deposits over the four-year period:

  

Concept used:

The amount after a year is given as the product of the principle (value of the deposit) at the start of the year and the growth factor.

Calculation:

The value of the second deposit for year 2 is $1000$ .

The growth factor is $g$ .

Then, the value of the second deposit for year 3 is $1000g$ and for year 4 is $1000g^2$ .

The value of the third deposit for year 3 is $1000$ .

The growth factor is $g$ .

Then, the value of the third deposit for year 4 is $1000g$ .

These are the required values that complete the given table.

Conclusion:

It has been determined that the value of the second deposit for year 3 is $1000g$ , its value for year 4 is $1000g^2$ and the value of the third deposit for year 4 is $1000g$ . These values complete the given table.

(b.)

To Write: A polynomial function that gives the value $v$ of the account at the end of the fourth summer in terms of $g$ .

It has been determined that a polynomial function that gives the value $v$ of the account at the end of the fourth summer in terms of $g$ , is $v=1000g^3+1000g^2+1000g+1000$ .

Given:

An amount of $\$1000$ is saved and deposited into a bank account at the end of each summer, each year towards buying a used car in four years.

The growth factor $1+r$ , where $r$ is the annual interest rate expressed as a decimal, is denoted by $g$ .

The following table shows the value of the deposits over the four-year period:

  

Concept used:

The value of the account at the end of the fourth summer is the sum of the values of all the deposits at the end of the fourth summer.

Calculation:

According to the completed table obtained in part (a), the value of the first deposit at the end of the fourth summer is $1000g^3$ , the value of the second deposit at the end of the fourth summer is $1000g^2$ , the value of the third deposit at the end of the fourth summer is $1000g$ , and lastly, the value of the fourth deposit at the end of the fourth summer is $1000$ .

Then, the value $v$ of the account at the end of the fourth summer in terms of $g$ , is given as,

  $v=1000g^3+1000g^2+1000g+1000$

Conclusion:

It has been determined that a polynomial function that gives the value $v$ of the account at the end of the fourth summer in terms of $g$ , is $v=1000g^3+1000g^2+1000g+1000$ .

(c.)

The growth factor and the annual interest rate needed to obtain the given amount.

It has been determined that the required growth factor is approximately $1.048$ and the required annual interest rate is approximately $4.8\%$ .

Given:

An amount of $\$1000$ is saved and deposited into a bank account at the end of each summer, each year towards buying a used car in four years.

The growth factor $1+r$ , where $r$ is the annual interest rate expressed as a decimal, is denoted by $g$ .

The following table shows the value of the deposits over the four-year period:

  

The amount to be saved is $\$4300$ .

Concept used:

The required growth factor and thus the required annual interest rate can be obtained by plugging in the given value of $v$ and determining the roots of the resulting polynomial.

Calculation:

As determined previously, the value $v$ of the account at the end of the fourth summer in terms of $g$ , is given as,

  $v=1000g^3+1000g^2+1000g+1000$

It is given that the amount to be saved is $\$4300$ .

Then, $v=4300$ .

Put $v=4300$ in the above polynomial function to get,

  $1000g^3+1000g^2+1000g+1000=4300$

Simplifying,

  $g^3+g^2+g+1=4.3$

On further simplification,

  $g^3+g^2+g+1-4.3=0$

Finally,

  $g^3+g^2+g-3.3=0$

The graph of the above polynomial is given as,

  

From the above graph, the only real root of the polynomial equation is $g\approx 1.048$ .

Hence, this is the required growth factor.

Now, as assumed,

  $1+r=g$

Put $g\approx 1.048$ in $1+r=g$ to get,

  $1+r\approx 1.048$

Solving,

  $r\approx 0.048$

This implies that the required annual interest rate is approximately $4.8\%$ .

Conclusion:

It has been determined that the required growth factor is approximately $1.048$ and the required annual interest rate is approximately $4.8\%$ .