A sample of a radioactive substance decayed to 91.5% of its original amount after a year. What is the half life of the substance? Also have to find the amount of time for it to decay to 30% of the original amount.
Radioactivity can be defined using this model:
$\displaystyle N=N_0e^{-kt}$, where
$\displaystyle N$ is the current amount of the substance
$\displaystyle N_0$ is the initial amount of the substance
$\displaystyle k$ is the decay constant
$\displaystyle t$ is the length of time
Now, we are told that we have $\displaystyle .915N_0$ after 1 year.
Thus, $\displaystyle .915N_0=N_0e^{-k}$
Solving for k, we see that $\displaystyle k=-\ln(.915)\approx 0.0888$
Now, we can find the half life:
Half life is defined as $\displaystyle \lambda=\frac{\ln 2}{k}$, where
$\displaystyle \lambda$ is the half life
$\displaystyle k$ is the decay constant
Thus, $\displaystyle \lambda=\frac{\ln 2}{.0888}\approx \color{red}\boxed{7.803\text{ years}}$
Now, our decay model is $\displaystyle N=N_0e^{-0.0888t}$
Can you find $\displaystyle t$ when $\displaystyle N=.3N_0$?
--Chris