Let N(t) be the fraction of the population who have heard a given piece of news t hours after its initial release. According to one model, the rate N'(t) at which the news spreads is equal to k times the fraction of the population that has not yet heard the news, for some constant k > 0.

(a) Determine the differential equation satisfied by N(t).

(b) Find the solution of this differential equation with the initial condition N(0) = 0 in terms of k.

(c) Suppose that half of the population is aware of an earthquake 8 hours after it occurs. Use the model to calculate k and estimate the percentage that will know about the earthquake 12 h after it occurs.