1. ## 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.

2. $.9A = Ae^{-\frac{t_1}{r}}$

$.9 = e^{-\frac{t_1}{r}}$

$\ln(.9) = -\frac{t_1}{r}$

$t_1 = -\frac{\ln(.9)}{r}$

using the same algebra ...

$t_2 = -\frac{\ln(.1)}{r}$

$\Delta t = t_2 - t_1 = -\frac{\ln(.1)}{r} + \frac{\ln(.9)}{r}$

$\Delta t = \frac{\ln(9)}{r}$