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

The time it will take for this investment is;

`t=19.50\text{ years}`

Step-by-step 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+(r)/(n))^(nt)`

Where t is the time of investment.

Making t the subject of the formula;

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

Substituting the given values;

`\begin{gathered} t=(\ln((F)/(P)))/(n\ln(1+(r)/(n)))=(\ln ((27696)/(6000)))/(2\ln (1+(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}`