Sociologists sometimes use the phrase “social diffusion” to describe the way information spreads through a population. The information might be a rumor, a cultural fad, or news about a technical innovation. In a sufficiently large population, the number of people x who have the information is treated as a differentiable function of time t, and the rate of diffusion, dx/dt, is assumed to be proportional to the number of people who have the information times the number of people who do not. This leads to the equation

where N is the number of people in the population. Suppose t is in days, k = 1/250, and two people start a rumor at time t = 0 in a population of N = 1000 people.

a. Find x as a function of t.

b. When will half the population have heard the rumor?

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dx dt = kx(N-x),