2021 ACJC CA2 Q3

A study is conducted by the National Environment Agency (NEA) to investigate the mosquito population growth at a breeding site. NEA found that at time \(t\) days after the study began, the birth rate of the mosquitoes is proportional to the number of mosquitoes, \(x\) thousands, present. Additionally, the death rate of the mosquitoes is proportional to the square of the number of mosquitoes present at time \(t\) days.

Initially, there are \(100\) mosquitoes at the breeding site. When \(x=2\), the number of mosquitoes remains constant.

1. Show that \(\frac{\mathrm{d}x}{\mathrm{d}t}=kx\left( 2-x \right)\), where \(k\) is a positive constant.[2]
2. The scientist found out the number of mosquitoes has doubled after \(30\) days. Show that \(x=\frac{2}{1+A{{\mathrm{e}}^{-Bt}}}\), where \(A\) and \(B\) are constants to be determined.[5]
3. Sketch the graph of \(x\) against \(t\).[2]