If $P is invested for n years at 12% compounded continuously, the rate at which the future value is growing is . In how many years will the future value double? Round your answer to one decimal place.
A)9.2 years
B)7.8 years
C)5.8 years
D)11.2 years
E)10.8 years

Question

If $P is invested for n years at 12% compounded continuously, the rate at which the future value is growing is  . In how many years will the future value double? Round your answer to one decimal place. ​
A)9.2 years 
B)7.8 years 
C)5.8 years 
D)11.2 years 
E)10.8 years

Quizplus · Accepted Answer

The formula for continuous compounding is given by $A=P\cdot e^{rt}$, where $A$ is the future value, $P$ is the principal, $r$ is the interest rate, and $t$ is the time in years. Taking the derivative of $A$ with respect to $t$, we get:

$\frac{dA}{dt}=rPe^{rt}$

We are given that $\frac{dA}{dt}=11eb0d3b_988f_bb40_9790_0f16398048df_TB1243_11$, and $r=0.12$. Therefore:

$11eb0d3b_988f_bb40_9790_0f16398048df_TB1243_11=0.12Pe^{0.12t}$

Solving for $t$:

$t=\frac{1}{0.12}\ln\left(\frac{11eb0d3b_988f_bb40_9790_0f16398048df_TB1243_11}{0.12P}\right)\approx 5.8$

So it will take approximately 5.8 years for the future value to double. Therefore, the correct answer is (C).

If $P Is Invested for N Years at 12% Compounded