2025 TMJC P2 Q4

A viral trend on social media gains popularity rapidly but eventually slows down due to oversaturation. Let \(P\) represent the popularity score of the trend at time \(t\) (measured in days), where \(0 \le t \le 100\). The change in the trend's momentum is modelled by \(\frac{\mathrm{d}^2P}{\mathrm{d}t^2} = -0.05\left(\frac{\mathrm{d}P}{\mathrm{d}t}\right)^2.\)

1. By substituting \(v = \frac{\mathrm{d}P}{\mathrm{d}t}\), show that \(v = \frac{20}{t + C}\) where \(C\) is a real constant.[3]
2. Given that \(P = 0\) and \(v = 10\) when \(t = 0\), find the particular solution of \(P\) in terms of \(t\).[4]
3. Find the popularity score of the trend when \(t = 30\).[1]