2023 SJI P2 Q2

The population of a type of virus at the end of \(t\) days can be modelled by the function

\[\mathrm P(t)=(2.5\times10^{15})(2\times10^{-6})^{kt},\quad\text{where}\hspace{0.5em}k<0\text{ and}\hspace{0.5em}t\geq0.\]

It is known that the virus population triples at the end of the first \(20\) days. Find the value of \(k\) and use this model to estimate the population of this virus at the end of \(1000\) days.[5]