2023 CJC P2 Q3

Following the popularity of the action role-playing game, Ginseng Impact, three years ago, developers have developed a strategy game, Ginseng Impactful. The number of people who download Ginseng Impactful, \(P\) (in thousands), in a particular city, at time \(t\) months, can be modelled by the differential equation

\(\frac{\mathrm{d}P}{\mathrm{d}t}=\frac{1}{26}P\left( 13-2P \right)\).

There were \(2000\) people who download Ginseng Impactful when it is launched.

1. Show that \(P=\frac{26}{9{{e}^{-\frac{1}{2}t}}+4}\).[6]
2. Determine, the time taken, in months, for the number of people who download Ginseng Impactful to double since the launch.[2]
3. Find the number of people that download Ginseng Impactful in the long run.[2]
4. Hence sketch the graph showing the number of people that download Ginseng Impactful against time.[2]