In a sufficiently large population, the number of people x who have a certain piece of information can be treated as a differentiable function of time t, and the rate at which th dx information spreads, is assumed to be proportional to the number of people who 1 dt have the information times the number of people who do not. This leads to the equation following equation dx - = kx(N − x), where N is the number of people in the population. dt 1 Suppose t is in days, k = and two people share the information at time t = 0 in a 250 population of N = 500 people. (a) Find x as a function of t. (b) When will half the population have heard the information? (This is when the rumor will be spreading the fastest.)

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