Finding half life given decay rate

Hey guys, I'm working on a problem right now and I'm having some trouble.

The decay of a certain chemical is 9.3% per year. Using the exponential decay model P(t) = P_{0}^{-kt d} where k is the decay rate, and P_{0 }is the original amount of chemical find the half-life.

Thanks!

Re: Finding half life given decay rate

Just out of interest did you get the answer 7.1 years?

Re: Finding half life given decay rate

Yes that is the correct answer.

Re: Finding half life given decay rate

I'm a bit surprized- that formula doesn't make sense. If $\displaystyle P(t)= P_0^{kt d}$, then the "initial value" is $\displaystyle P(0)= 1$, not $\displaystyle P_0$. Did you mean $\displaystyle P(t)= P_0e^{kt d}$? And if $\displaystyle P_0$ is the initial value, t is the time, and k is the decay rate, what is "d"?

More common is the formula $\displaystyle P(t)= P_0e^{kt}$ for decay. In one year, we will have $\displaystyle P(1)= P_0e^k$. If the chemical decays 9.3% per year, P(1) should equal $\displaystyle (1- .093)P_0= .907P_0$ so $\displaystyle P_0e^k= .907P_0$. That is, k must satisfy $\displaystyle e^k= .907$ so k= ln(.907)= -.0976.

Once you know that, the "half life" is the time until $\displaystyle P_0$ becomes $\displaystyle P_0/2$. Solve $\displaystyle P_0e^{-.0976t}= P_0/2$ which is the same as $\displaystyle e^{-.0976t}= 1/2$. Solve that for t.