1380. The rate of decay of radium is proportional to its current amount $R$. Find the dependence of $R$ on $t$; formulate the differential equation and determine the proportionality coefficient from experimental data, according to which after 1600 years half of the current amount of radium will remain.
Solution. According to the problem, $\frac{d R}{d t}=k R$ (the rate of change is the derivative with respect to time), from which $\frac{d R}{R}=k d t$,

$$
\begin{gathered}
\ln R=k t+\ln C, \quad \ln R=\ln e^{k t}+\ln C= \\
=\ln C e^{k t}, \quad R=C e^{k t} .
\end{gathered}
$$

Let the amount of radium at the initial moment of time $t=0$ be $R=R_{0}$, then $R_{0}=C e^{k 0}$, from which $C=R_{0}$. Substituting this value of $C$ into expression (b), we get

$$
R=R_{0} e^{k t}
$$

The coefficient $k$ is found from the condition that if $t=1600$, then $R=\frac{R_{0}}{2}$. Substituting these values into formula (g), we get

$$
\frac{R_{0}}{2}=R_{0} e^{1600 k}
$$

from which $\frac{1}{2}=e^{1600 k}, \quad-\ln 2=1600 k, \quad k=\frac{-\ln 2}{1600} \approx-0.00043$. Therefore, the desired function is $R=R_{0} e^{-0.00043 t}$.