Chapter 3.6, Problem 5E

Solution

To find: The amount $A$ accumulated after investing a principal $P$ for $t$ years at an interest rate $r$ compounded $k$ times per year.

The amount is $\$46,057.58$ .

Given information:

It is given that $P=\$1500,r=7\%,t=5,k=4$ .

Concept Used:

Interest Compounded $k$times per year If a principal $P$ is invested at a annual interest rate $r$ , calculated $k$ times a year for $t$ years, then $\frac{r}{k}$ is the interest rate per compounding period, and $kt$ is the number of compounding periods. The amount $A$ in the account after $t$ years is $A=P{\left(1+\frac{r}{k}\right)}^{kt}$ .

Calculation:

Here, $P=\$1500,r=7\%,t=5,k=4$

Substitute $P$ for $\$1500$ , $r$ for $0.07$ , $t$ for $5$ , and $k$ for $4$ in $A=P{\left(1+\frac{r}{k}\right)}^{kt}$ .

  $\begin{array}{c}A=1500{\left(1+\frac{0.07}{4}\right)}^{4(5)}\\ =1500{\left(\frac{4.07}{4}\right)}^{20}\\ =1500{\left(1.0175\right)}^{20}\\ =\$2122.17\end{array}$

Hence, amount $A$ accumulated after investing a principal $P$ for $t$ years at an interest rate $r$ compounded $k$ times per year is $\$2122.17$ .