Chapter 1.3, Problem 27E

To determine

To find: The time taken for an investment to double in value at the rate of $6.25\%$ compounded continuously.

Answer to Problem 27E

The time taken for an investment to double in value at the rate of $6.25\%$ compounded continuously is $11.09$ years.

Explanation of Solution

Given information: A certain amount is invested at the rate of $6.25\%$ compounded continuously.

Calculation:

Assume that initial investment is $\$P$ and the interest rate is $6.25\%$ compounded continuously.

The exponential function for the amount is given by:

  $y=P{e}^{\frac{rt}{100}}$

Substitute $2P$ for $y$ , $6.25$ for $r$ in the exponential formula.

  $\begin{array}{c}y=P{e}^{\frac{rt}{100}}\\ 2P=P{e}^{\frac{6.25t}{100}}\\ 2=e^{0.0625t}\end{array}$

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 $e^{0.0625t}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{LN}}\boxed{0.0625}\boxed{\text{X}}$ . Now, press the key $\boxed{\text{ENTER}}$ .
• Enter left hand side of the equation $2$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{2}$ .
• Press $\boxed{\text{WINDOW}}$ key and adjust the window at $\left[-10,20\right]$ by $\left[-10,20\right]$ .
• 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 $11.0903$ years.

Therefore, the time taken for an investment to double in value at the rate of $6.25\%$ compounded continuously is $11.09$ years.