Chapter 6.5, Problem 49E

To determine

To write and solve an exponential equation to determine when the balance of the account will be $\$800$ , if the deposit of $\$500$ in a savings account that earns $6\%$ annual interest compounded yearly.

Answer to Problem 49E

  $S(n)=500{\left(1+\frac{6}{100}\right)}^n$ is the required exponential equation that represents the amount in saving account after $n\text{years}$ and after $8.07\text{years}$ the deposited amount (principal amount) becomes $\$800$ .

Explanation of Solution

Given information :

Given the deposit of $\$500$ in a savings account that earns $6\%$ annual interest compounded yearly.

Since we have principal amount $P=\$500$ and that earns $r=6\%$ annual interest compounded yearly.

Interest amount after first year $rP$

So after first year, the amount in saving account is $S(1)=P+rP=P\left(1+r\right)$

Similar way after second year, the amount in saving account is $S(2)=P{\left(1+r\right)}^2$

In general after $n\text{years}$ , the amount in saving account is $S(n)=P{\left(1+r\right)}^n\mathrm{........}(i)$

Since $P=\$500$ and $r=6\%=\frac{6}{100}$

Thus, $S(n)=500{\left(1+\frac{6}{100}\right)}^n$ is the required exponential equation that represents the amount in saving account after $n\text{years}$ .

To determine after which year $(n)$ :

The deposited amount (principal amount) becomes $\$800$ , substitute $S(n)=800$ in equation $S(n)=500{\left(1+\frac{6}{100}\right)}^n$ , we get

  $\begin{array}{l}S(n)=500{\left(1+\frac{6}{100}\right)}^n\\ \Rightarrow 800=500{\left(1+\frac{6}{100}\right)}^n\\ \Rightarrow 8=5{\left(1+0.06\right)}^n\\ \Rightarrow \frac{8}{5}={\left(1.06\right)}^n\\ \Rightarrow \ln 1.6=\ln {\left(1.06\right)}^n\\ \Rightarrow \ln 1.6=\left(n\right)\ln \left(1.06\right)\\ \therefore n=\frac{\ln 1.6}{\ln \left(1.06\right)}\simeq 8.07\text{years}\end{array}$

Therefore, after $8.07\text{years}$ the deposited amount (principal amount) becomes $\$800$ .

Hence, $S(n)=500{\left(1+\frac{6}{100}\right)}^n$ is the required exponential equation that represents the amount in saving account after $n\text{years}$ and after $8.07\text{years}$ the deposited amount (principal amount) becomes $\$800$ .