2012 MJC P1 Q10

There was an island where initially there was no one living on it. The total capacity of the island is \(9000\). The population increases at a rate which is inversely proportional to the remaining capacity of the island. At the same time, the rate at which the population decreases is \(\frac{1}{20}\) of the population size. When the population reaches \(4000\), it remains at this value. The population size (in thousands) is \(x\) at time \(t\) months, show that

\(\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{\left( x-4 \right)\left( x-5 \right)}{20\left( 9-x \right)}.\)

Find \(t\) in terms of \(x\). Hence find the time when the population reaches \(2000\).[7]