If you have an initial mass of 50kg for a radioactive sample, which has a half-life of 5000 years, how would you go about finding an equation that describes the mass of the sample over time??
My approach:
N(t)=$N_0$$e^{-kt}$
N(5000)=50$e^{-5000k}$
I believe my approach to the problem is over simplified. Would the approach involve taking the derivative of $\frac{dN}{dt}$=$-kN$?
On
You have a half life. So at $t=5000$, the sample should have halved. You have the form of solution $N(t)=N_0 e^{-kt}$, and you know its mass at times $t=0$, and $t=5000$ - it is $50kg$ and $25kg$ respectively. Substituting this in gives two equations with 2 unknowns which can be solved to find $N_0$ and $k$.
Can you take it from here?
Really $N(t)=50\text{kg}\cdot 2^{-\frac{t}{5000\text{yr}}}$.
The formula $N(t)=N_0e^{-kt}$ is a correct formula where $k$ is the decay rate, but you are given half-time, and the decay rate is linked to the half-time via equation $t_{1/2}=\frac{\ln 2}{k}$ or $k=\frac{\ln 2}{t_{1/2}}$ (see https://en.wikipedia.org/wiki/Exponential_decay#Half-life).
Indeed, if you replace $k=\frac{\ln 2}{5000\text{yr}}$ you get:
$$N(t)=N_0e^{-\frac{\ln 2}{5000\text{yr}}t}=50\text{kg}\cdot 2^{-\frac{t}{5000\text{yr}}}$$