Question

6) How long will it take for an investment to double in value if it earns 4.75% compoundedcontinuously?A) 14.593 yearsB) 15.711 years C) 23.129 years D) 7.296 years

Answer 1

A quantity that is compounded continuously follows the next equation:

$A=Pe^{rt}$

Where:

$\begin{gathered} A=\text{ amount in time ''t''} \\ P=\text{ initial amount} \\ r=\text{ rate of interest in decimal form} \\ t=\text{ time} \end{gathered}$

Now, the interest rate in decimal notation is determined by dividing the percentage by 100:

$r=\frac{4.75}{100}=0.0475$

Now, we are asked to determine the time required for the quantity to double. Therefore, we need to determine "t" when:

$A=2P$

Substituting in the formula we get:

$2P=Pe^{rt}$

Now, we can cancel out the "P":

$2=e^{rt}$

Now, we solve for "t". First, we take the natural logarithm to both sides:

$ln2=lne^{rt}$

Now, we use the following property of logarithms:

$lnx{}^y=ylnx$

Applying the property we get:

$ln2=rtlne$

We have that:

$lne=1$

Therefore:

$ln2=rt$

Now, we divide both sides by "r":

$\frac{ln2}{r}=t$

Now, we substitute the value of "r":

$\frac{ln2}{0.0475}=t$

Solving the operations:

$14.593=t$

Therefore, the right option is A.