Answer
$$
R=A S^{k} \quad\quad [\text { R as a function of S.}]
$$
where $A=e^{C}$ is a positive constant.
Work Step by Step
$$
\frac{1}{R} \frac{d R}{d t}=\frac{k}{S} \frac{d S}{d t} \Rightarrow \frac{d}{d t}(\ln R)=\frac{d}{d t}(k \ln S)
$$
$
\Rightarrow
$
$$
\ln R=k \ln S+C \Rightarrow R=e^{k \ln S+C} = e^{C}\left(e^{\ln S}\right)^{k}
$$
$$
\Rightarrow R=A S^{k} \quad\quad [\text { R as a function of S.}]
$$
where $A=e^{C}$ is a positive constant.
