If $9,000 is invested at an annual rate of 11% per year compounded monthly, the future value S at any time t (in months) is given by . How long does it take for the investment to double? Round your answer to one decimal place.
A)91.2 months
B)14.8 months
C)76.0 months
D)92.2 months
E)68.7 months

Question

If $9,000 is invested at an annual rate of 11% per year compounded monthly, the future value S at any time t (in months) is given by  . How long does it take for the investment to double? Round your answer to one decimal place. ​
A)91.2 months 
B)14.8 months 
C)76.0 months 
D)92.2 months 
E)68.7 months

Quizplus · Accepted Answer

To find the time it takes for the investment to double, we need to solve the equation:
$2S=S_0e^{0.11t}$ where $S_0=9000$ and $t$ is the time in months.
Dividing both sides by $S_0$ and taking the natural logarithm gives:
$\ln 2 = \ln\left(e^{0.11t}ight)=0.11t\ln(e)=0.11t$
Solving for $t$ gives:
$t=\frac{\ln 2}{0.11}=6.3	ext{ years}=6.3	imes 12	ext{ months}=75.6	ext{ months}$.
Rounding to one decimal place, we get $t\approx \boxed{	extbf{(C)}\ 76.0\ 	ext{months}}$.

If $9,000 Is Invested at an Annual Rate of 11