2016 Specimen Paper P1 Q11 [RI Differential Equations Tutorial Q7]

A pond has a surface area of \(10\mathrm{ }{{\mathrm{m}}^{2}}\). Biologists have planted an area of new weeds. They estimate how many weeds there are and the rate at which they are spreading by finding the area of the pond the weeds cover at various times. They believe that the area \(A\mathrm{ }{{\mathrm{m}}^{2}}\) of weeds present at time \(t\) months is such that the rate at which the area is increasing is proportional to the product of the area of the pond covered by the weeds and the area of the pond not covered by the weeds. It is known that the initial area of weeds is \(2\mathrm{ }{{\mathrm{m}}^{2}}\) and that the area of weeds is \(4\)m\(^{2}\) after \(5\) months.

1. Write down a differential equation expressing the relation between \(A\) and \(t\). Find the time at which \(80\%\) of the pond is covered in weeds, giving your answer correct to \(2\) decimal places.[8]
2. Given that the experiment is stopped after \(2\) years, find the area of pond covered by weeds, giving your answer correct to \(2\) decimal places.[2]
3. Write the solution of the differential equation in the form \(A=\mathrm{f}\left( t \right)\) and sketch this curve.[4]