The question was:
A radioactive element decays at a rate proportional to the mass remaining. Initially the mass is 10mg and after 20 days it is 5mg. Set up a differential equation describing this situation and solve it to find the time taken to reach 1mg.
Can someone please tell me how do I find dy/dt at t=0 because I can't seem to get that from the question although I tried solving it by diving 5 by 20 it get the rate of change per day, but I ended up with t equals to a negative number so there must be something wrong.
Any ideas?
Best Answer
The differential equation that describes the mass of a radioactive element is
$$\frac{dy}{dt}=-ky,$$ where $k$ is some constant. If you solve the differential equation then you get that
$$y(t)=y(0)e^{-kt}.$$ Since $y(0)$ it is assumed to be $10$ $\text{mg}$ then it is
$$y(t)=10e^{-kt}.$$
Now, since it is $$5=10e^{-20k}$$ we get that $k=\frac{\ln 2}{20}.$ Thus $y'(0)=-10k=-\ln \sqrt{2}.$