Chapter 1.3, Problem 24E

To determine

To find: The time taken for the John’s balance to reach $\$4150$ .

Answer to Problem 24E

The time taken for the John’s balance to reach $\$4150$ is approximate $11$ years.

Explanation of Solution

Given information: John invests an amount of $\$2300$ in saving account with a $6\%$ interest rate compounded annually. The total balance of the account will be $\$4150$ .

Calculation:

The initial investment is $\$2300$ and the interest rate is $6\%$ compounded annually.

The formula for the value is:

  $y=P{\left(1+\frac{r}{100}\right)}^n$

Substitute $\$2300$ for $P$ , $6\%$ for $r$ and $\$4150$ for $y$ in the formula.

  $\begin{array}{c}y=P{\left(1+\frac{r}{100}\right)}^n\\ 4150=2300{\left(1+\frac{6}{100}\right)}^n\\ 4150=2300{\left(1+0.06\right)}^n\\ 4150=2300{\left(1.06\right)}^n\end{array}$

Follow the steps to solve the equation by calculator:

• First press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key.
• Enter right hand side of the equation $2300{\left(1.06\right)}^n$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2300}\boxed{\times }\boxed{(}\boxed{1.06}\boxed{)}\boxed{^}\boxed{\text{X}}$ . Now, press the key $\boxed{\text{ENTER}}$ .
• Enter left hand side of the equation $4150$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{4150}$ .
• Press $\boxed{\text{WINDOW}}$ key and adjust the window at $\left[-20,20\right]$ by $\left[-100,5000\right]$ .
• Now, press keystrokes given below to find the value of $t$ as:

  $\boxed{2\text{nd}}\to \boxed{\text{TRACE}}\to \boxed{5}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}$

  

So, the value of t is $10.12$ years.

Therefore, the time taken for the John’s balance to reach $\$4150$ is approximate $11$ years.