2025 CJC P1 Q9

A Koi pond farmer introduces a circular patch of duckweed at the center of his circular pond to enhance water purification and provide supplemental nutrition for his Koi, as shown in the figure below.

The duckweed grows over time \(t\) days, forming a continuous circular patch on the pond's surface. It is assumed that the duckweed forms a thin, floating layer with negligible thickness. The rate of increase of the area covered by duckweed is directly proportional to the area of the pond that remains uncovered by the duckweed.

1. Given that the radius of the pond is \(10\) m, show that the radius, \(r\) m of the circular patch of duckweed satisfies the differential equation \(\frac{\mathrm{d}r}{\mathrm{d}t} = \frac{k(100 - r^2)}{2r}\), \(k > 0\).[2]
2. The initial radius of the circular patch of duckweed is \(1\) m. Solve the differential equation, expressing \(r\) in the form \(r = \sqrt{P - Qe^{-kt}}\), where \(P\) and \(Q\) are constants to be determined.[6]
3. Sketch the graph of \(r\) against \(t\).[3]