Chapter 7.7, Problem 28E

Solution

a.

To calculate: the balance after end of second year and third year.

Time of simple interest is $5\text{years}$

Given information:

The amount deposited $\$3000$ at the start of every year that earns $7\%$ interest compounded annually.

Concept Used:

Compound interest problem requires the use of the following formula:

  $A=P{(1+\frac{r}{n})}^{nt}$

Calculation:

As per the given problem

  $\text{Principal(}P)=\$3000,\text{interest}\text{rate(r)=7\% ,t=2 year and 3 year ,and n=1}$

Therefore, the balance in the account after $2\text{year}$ obtained as:

Substitute the values in the formula

  $A=P{(1+\frac{r}{n})}^{nt}$

  $\Rightarrow A=3000{(1+\frac{.07}{1})}^{1\times 2}$

  $\Rightarrow A=3000{(1.07)}^2$

  $\Rightarrow A=3434.7$

The balance in the account after $2\text{year}$ is $\$3434.70$

And , the balance in the account after $\text{3year}$ obtained as:

  $A=P{(1+\frac{r}{n})}^{nt}$

  $\Rightarrow A=3000{(1+\frac{.07}{1})}^{1\times 3}$

  $\Rightarrow A=3000{(1.07)}^3$

  $\Rightarrow A=3675.129$

The balance in the account after $2\text{year}$ is $\$3675.129$

Hence,

The balance at the end of the second year and third year is $\$3434.70$ and $\$3675.129$ respectively.

Conclusion:

The balance at the end of the second year and third year is $\$3434.70$ and $\$3675.129$ respectively.

b.

To explain: money at the end of the fifth year to buy a car of costs $\$18,000$ is enough or not.

The cost of car is $\$18,000$ so amount earned in five year is $\$4207.65$ which is not sufficient to buy the car.

Given information: The amount deposited $\$3000$ at the start of every year that earns $7\%$ interest compounded annually.

Concept Used:

Compound interest problem requires the use of the following formula:

  $A=P{(1+\frac{r}{n})}^{nt}$

Calculation:

As per the given problem,

  $\text{Principal(}P)=\$3000,\text{interest}\text{rate(r)=7\% ,t=5 year, and n=1}$

Substitute the values in the formula

  $A=P{(1+\frac{r}{n})}^{nt}$

  $\Rightarrow A=3000{(1+\frac{.07}{1})}^{1\times 5}$

  $\Rightarrow A=3000{(1.07)}^5$

  $\Rightarrow A=4207.655$

As, the cost of car is $\$18,000$ so amount earned is $\$4207.65$ which is not sufficient to buy the car.

Conclusion:

The cost of car is $\$18,000$ so amount earned in five year is

  $\$4207.65$ which is not sufficient to buy the car.