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.

2. Originally Posted by GreenandGold
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:

$N=N_0e^{-kt}$, where

$N$ is the current amount of the substance
$N_0$ is the initial amount of the substance
$k$ is the decay constant
$t$ is the length of time

Now, we are told that we have $.915N_0$ after 1 year.

Thus, $.915N_0=N_0e^{-k}$

Solving for k, we see that $k=-\ln(.915)\approx 0.0888$

Now, we can find the half life:

Half life is defined as $\lambda=\frac{\ln 2}{k}$, where

$\lambda$ is the half life
$k$ is the decay constant

Thus, $\lambda=\frac{\ln 2}{.0888}\approx \color{red}\boxed{7.803\text{ years}}$

Now, our decay model is $N=N_0e^{-0.0888t}$

Can you find $t$ when $N=.3N_0$?

--Chris

3. take the natural log of both sides?

4. Originally Posted by GreenandGold
take the natural log of both sides?

--Chris

5. I got -13 but i did something wrong. is it 7.803(ln(.3)/ln(2))

6. ok i added a negative to the decay since its negative. so i got 1.35535

7. Originally Posted by GreenandGold
ok i added a negative to the decay since its negative. so i got 1.35535
Hmmm...

$N=N_0e^{-0.0888t}\implies.3N_0=N_0e^{-0.0888t}$ $\implies \ln(.3)=-0.0888t\implies t=-\frac{\ln(.3)}{0.0888}\approx\color{red}\boxed{13. 558\text{ years}}$

--Chris

8. ok i see now!!! thanks...