Question: Techniques of Integration I

Sociologists sometimes use the phrase social diffusion to describe the way information spreads through a population. 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, , 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, , and two people start a rumor at time in a population of N=1000 people.

a) Find a as a function of t.

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

Given the above problem, provide written responses to address the following points as precisely and thoroughly as possible.

A. Explain the problem in your own words.

B. What mathematical concepts learned in this module apply to this problem?

C. Explain the steps you must take to solve this problem.

D. What is the most difficult aspect of solving this problem?

E. Explain exactly what the answer means from a mathematical perspective.