Applications and ExamplesSocial diffusion Sociologists

Chapter 8, Problem 59E

(choose chapter or problem)

Problem 59E

Applications and Examples

Social diffusion 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? (This is when the rumor will be spreading the fastest.)