Question

How long will it take for a $6000 investment to grow to $27,696 at an annual rate of 8%, compounded semiannually? Assume that no withdrawals aremade.Do not round any intermediate computations, and round your answer to the nearest hundredth.Please put final answers in years

Answer 1

Answer:

The time it will take for this investment is;

$t=19.50\text{ years}$

Explanation:

Given that the initial investment was $6,000.

$P=6,000$

And the final amount is $27,696.

$F=27,696$

at an annual rate of 8%, compounded semiannually;

$\begin{gathered} r=0.08 \\ n=2 \end{gathered}$

Recall that the formula for calculating compound interest can be written as;

$F=P(1+\frac{r}{n})^{nt}$

Where t is the time of investment.

Making t the subject of the formula;

$\begin{gathered} F=P(1+\frac{r}{n})^{nt} \\ \frac{F}{P}=(1+\frac{r}{n})^{nt} \\ \ln (\frac{F}{P})=nt\ln (1+\frac{r}{n}) \\ t=\frac{\ln(\frac{F}{P})}{n\ln(1+\frac{r}{n})} \end{gathered}$

Substituting the given values;

$\begin{gathered} t=\frac{\ln(\frac{F}{P})}{n\ln(1+\frac{r}{n})}=\frac{\ln (\frac{27696}{6000})}{2\ln (1+\frac{0.08}{2})} \\ t=19.50\text{ years} \end{gathered}$

Therefore, the time it will take for this investment is;

$t=19.50\text{ years}$