N2025 P1 Q6

The carbon content of plants includes a small proportion of the radioactive isotope carbon-14. The mass of carbon-14 in dead plants decreases over time. Scientists measure the mass of carbon-14 in plants to estimate how long ago they died.

The mass of carbon-14 in dead plants decreases by \(50\%\) in \(5730\) years.

The mass, \(M\), of carbon-14 in a plant \(t\) years after it has died is modelled by the differential equation

\[\frac{\mathrm{d}M}{\mathrm{d}t} = -kM,\]where \(k\) is a positive constant. The initial value of \(M\) is \(M_0\).

1. By solving the differential equation, find the value of \(k\).[4]

2. A dead plant has \(20\%\) of its original mass of carbon-14 remaining. Determine, to the nearest \(100\) years, how long ago the plant died.

   [2]