# Continuous compounding

• Mar 11th 2009, 11:39 AM
hotblonde
Continuous compounding
I'm having a lot of trouble with my calculus homework. Does anyone know how to work this problem?
Suppose the consumption of electricity grows at 7.6% per year, compounded continuously. Find the number of years before the use of electricity has tripled. Round the answer to the nearest hundredth.
http://blackboard.selu.edu/images/spacer.gif
14.46 , 39.47 , 0.14 , 1.45
• Mar 11th 2009, 02:02 PM
hotblonde
ANYBODY??? I figured the answer but am having problems with the formula. I came up with fv=p(1+r/n)^t, and cannot figure out t....my formula looks like this:

3=1(1.076)^t

help...........
• Mar 11th 2009, 03:04 PM
billa
I got 14.55 years - I will post how I did it in a minute - I have to get the equations into this box

a(t) = lim as n approaches infinity ( a0(1+ r/n)^(nt) )
where a0 is the initial amount of power used, r is the rate of growth (.076) and n is the number of times it is compounded

Then you use L'Hopital's rule to find that the above function is the same as this

a(t) = a0e^(rt)

(Hopefully this helps a little)

$\displaystyle \mathop {\lim }\limits_{n \to \infty } \ln ((1 + \frac{r} {n})^{nt} ) = \mathop {\lim }\limits_{n \to \infty } (nt)\ln (1 + \frac{r} {n}) = \mathop {\lim }\limits_{n \to \infty } t \bullet \frac{{\ln (1 + \frac{r} {n})}} {{1/n}}$

$\displaystyle \begin{gathered} = \mathop {\lim }\limits_{n \to \infty } t \bullet \frac{{\frac{1} {{1 + \frac{r} {n}}}}} {{ - 1/n^2 }}( - r/n^2 ) = \mathop {\lim }\limits_{n \to \infty } t\frac{n} {{n + r}}(r) = \frac{1} {1}rt \hfill \\ \mathop {\lim }\limits_{n \to \infty } e^{\ln f(x)} = e^L = e^{rt} \hfill \\ \end{gathered}$

a(t) = 3 = 1*e^(.076*t)
ln(3) = .076*t
t=ln(3)/.076 = 14.55 years
• Mar 11th 2009, 03:31 PM
ahhh
Anytime something is compounded continuously, use the $\displaystyle A=Ce^{kt}$ formula. Or, if it is money that is being compounded continuously, use $\displaystyle A=Pe^{rt}$ formula. Either way, the formulas are equivalent.

Or like billa said, you can use the regular compound formula and just find the $\displaystyle {\lim x \rightarrow \infty}$.