How would I solve the following word problem.
A radioactive substance weighed $n$ grams at time t=0. Today 5 years later the substance weight $m$ grams.How much will the substance weight 5 years from now?
How would I solve this I know the formula I have to use is $f(x)=Ce^{kt}$ and to find $k$ you do $\ln2$/half life
Best Answer
We don't need to calculate $k$. We have $$f(t)=Ce^{kt}.$$
We are told that $f(0)=n$, so $C=n$.
We are told that $f(5)=m$. So $$Ce^{5k}=ne^{5k}=m.\tag{1}$$
We want to find $f(10)$, which is $ne^{10k}$. But from (1) $e^{5k}=\frac{m}{n}$, so $e^{10k}=\frac{m^2}{n^2}$ and therefore $$f(10)=n\cdot \frac{m^2}{n^2}=\frac{m^2}{n}.$$