9758/2020/P1/Q10

Scientists are investigating the effect of disease on the number of sheep on a small island. They discover that every year the death rate of the sheep is greater than the birth rate of the sheep. The difference every year between the death rate and the birth rate for the population of sheep on the island is $3 \%$. The number of sheep on the island is $P$ at a time $t$ years after the scientists begin observations.

(i) Write down a differential equation relating $P$ and $t$.

[2]

(ii) Solve this differential equation to find an expression for $P$ in terms of $t$. Explain what happens to the number of sheep if this situation continues over many years.

[4]

The scientists import sheep at a constant uniform rate of $n$ sheep per year. (The difference every year between the death rate and the birth rate remains at $3 \%$.)

(iii) Write down a differential equation to model the new situation.

[2]

(iv) Solve the differential equation to find an expression for $P$ in terms of $t$ and $n$.

[4]

(v) Given that the number of sheep settles down to 500 after many years, find $n$.

[2]