Express $\frac{1}{Q(500 - Q)}$ in partial fractions.[2] A group of scientists is monitoring the po

Question

Express $\frac{1}{Q(500 - Q)}$ in partial fractions.[2] A group of scientists is monitoring the population of a particular species of birds on an island. At time $t$ years after the start of the monitoring, the number of birds on the island is $Q$. The scientists observe that the birth rate is proportional to the bird's population and the birds are dying at a rate proportional to the square of the bird's population. There were $1000$ birds on the island when the scientists first started monitoring the population. They discover that the population remains unchanged when there are $500$ birds. Show that the differential equation relating $Q$ and $t$ is given by$$\frac{\mathrm{d}Q}{\mathrm{d}t} = kQ(500 - Q)$$where $k$ is a positive constant.[2] Hence, solve the differential equation, expressing $Q$ in terms of $k$ and $t$.[5]

Accepted Answer

(a) $\dfrac1{Q(500-Q)}=\dfrac1{500Q}+\dfrac1{500(500-Q)}$ (b)(i) $\dfrac{\mathrm dQ}{\mathrm dt}=kQ(500-Q)$ (ii) $Q=\dfrac{1000}{2-e^{-500kt}}$

2025 HCI P1 Q10

2025 HCI P1 Q10

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