2022 ACJC P1 Q11

With the proliferation of online social network, sociologists recognised a phenomenon called social diffusion which is the spreading of a piece of information through the population.The members of the population can be classified into two categories namely those who have the information and those who do not.

It is given that \(N\) denotes the number of people, in thousands, who have the information in a fixed population size \(P\), in thousands. The rate of diffusion of a piece of information on social media, \(\frac{\mathrm{d}N}{\mathrm{d}t}\), where \(t\) represents the time taken in hours, can be modelled as being proportional, with a constant of proportionality \(k\), to the product of the number of people who have the information and the number of people who have yet to receive it.

1. Write down the differential equation involving \(N\), \(t\), \(k\) and \(P\).[1]

It is given that \(N=1\) and \(\frac{\mathrm{d}N}{\mathrm{d}t}=\frac{P-1}{10}\) when \(t=0\).

2. Show that the general solution of the differential equation in part (i) is \(N=\frac{AP}{A+{{\mathrm{e}}^{-0.1Pt}}}\), where \(A\) is an arbitrary constant.[6]
3. Given that the population size is \(1\) million, find the particular solution and hence sketch the graph of \(N\) against \(t\).[3]
4. With reference to the graph of \(N\) against \(t\), explain in context, the long-term implication of the model used.[1]
5. Find the time taken, to the nearest minute, for a piece of information to reach \(99\%\) of the population of one million.[1]