Sociologists recognize a phenomenon called social diffusion, which is the spreading of a piece of information,...

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Sociologists recognize a phenomenon called social diffusion, which is the spreading of a piece of information, a technological innovation, or a cultural fad among a population. The members of the population can be divided into two classes: those who have the information and those who do not. In a fixed population whose size is known, it is reasonable to assume that the rate of diffusion is proportional to the number who have the information times the number yet to receive it. If X denotes the number of individuals who have the information in a population of N people, then a mathematical model for social diffusion is given by dX/dt = kX(N – X), where t represents time and k is a positive constant. a. Solve the model and show that it leads to a logistic curve. b. At what time is the information spreading fastest? c. How many people will eventually receive the information? Sociologists recognize a phenomenon called social diffusion, which is the spreading of a piece of information, a technological innovation, or a cultural fad among a population. The members of the population can be divided into two classes: those who have the information and those who do not. In a fixed population whose size is known, it is reasonable to assume that the rate of diffusion is proportional to the number who have the information times the number yet to receive it. If X denotes the number of individuals who have the information in a population of N people, then a mathematical model for social diffusion is given by dX/dt = kX(N – X), where t represents time and k is a positive constant. a. Solve the model and show that it leads to a logistic curve. b. At what time is the information spreading fastest? c. How many people will eventually receive the information?