The formula for compound continuously rate [tex]A = P e^{rt} [/tex].
Where is A (future amount) = 2519, P (initial amount) = 2300, e is the mathematical constant, r (rate of interest) which is unknown, and t (time in years) = 20
Now we plug in the variables into the equation.
[tex]2519 = 2300 e^{r*20} [/tex]
It is asking for the rate, so must isolate [tex]r[/tex].
First we divide 2300 on both sides of the equation.
[tex] \frac{2519}{2300} = \frac{2300}{2300} e^{r*20}[/tex]
[tex]1.0952 = e^{(r*20)} [/tex]
Then we enter the inverse of [tex]e[/tex], which is log.
[tex]log(1.0952)=log(e^{r*20})[/tex]
[tex]0.0395=20r * log(e)[/tex]
Then divide [tex]log(e)[/tex] on both sides.
[tex] \frac{0.0395}{log(e)} =20r * \frac{log(e)}{log(e)} [/tex]
[tex]\frac{0.0395}{log(e)} =20r[/tex]
Then divide 20 on both sides.
[tex]\frac{0.0395}{20*log(e)} = \frac{20r}{20} [/tex]
[tex]\frac{0.0395}{20*log(e)} = r[/tex]
Finally solve for [tex]r[/tex]
0 = r
The answer is r = 0%
