9709 P31 - Nov 2019 - Q4
2319
The number of insects in a population \(t\) weeks after the start of observations is denoted by \(N\). The population is decreasing at a rate proportional to \(Ne^{-0.02t}\). The variables \(N\) and \(t\) are treated as continuous, and it is given that when \(t = 0\), \(N = 1000\) and \[ \frac{dN}{dt} = -10. \]
(i) Show that \(N\) and \(t\) satisfy the differential equation \[ \frac{dN}{dt} = -0.01e^{-0.02t}N. \]
(ii) Solve the differential equation and find the value of \(t\) when \(N = 800\).
(iii) State what happens to the value of \(N\) as \(t\) becomes large.
