Answer
5.5 disintegrations per minute.
Work Step by Step
Original activity $A_{0}=15.3\,dpm$
Decay constant $k=\frac{0.693}{t_{1/2}}=\frac{0.693}{5715\,y}=1.2126\times10^{-4}\,y^{-1}$
Time $t=8.4\times10^{3}\,y$
$\ln(\frac{A_{0}}{A})=kt$ where $A$ is the current activity.
$\implies \ln(\frac{15.3\,dpm}{A})=1.2126\times10^{-4}\,y^{-1}\times8.4\times10^{3}\,y=1.0186$
Taking the inverse $\ln$ of both sides, we have
$\frac{15.3\,dpm}{A}=e^{1.0186}=2.7693$
Or $A= \frac{15.3\,dpm}{2.7693}=5.5\,dpm$
