
Exponential Decay
A first order exponential decay can be written as
A(t)= Ae^(t/r)
where A(t) is the amount (of substance) after time t, A is the initial amount at time t=0 and r is the decay time.
The fall time is defined as the time in which A(t) falls from 90% to 10% of its initial value. Find the relationship between the fall time of an exponential decay and r.

$\displaystyle .9A = Ae^{\frac{t_1}{r}}$
$\displaystyle .9 = e^{\frac{t_1}{r}}$
$\displaystyle \ln(.9) = \frac{t_1}{r}$
$\displaystyle t_1 = \frac{\ln(.9)}{r}$
using the same algebra ...
$\displaystyle t_2 = \frac{\ln(.1)}{r}$
$\displaystyle \Delta t = t_2  t_1 = \frac{\ln(.1)}{r} + \frac{\ln(.9)}{r}$
$\displaystyle \Delta t = \frac{\ln(9)}{r}$