A rumor is spread when “information” passes from person to person. One assumption to understand how fast a rumor spreads is; people acquire the rumor by means of public sources, such as the Internet and the rate of acquisition is proportional to the number of people that don’t yet know the rumor.

(a) Develop a mathematical model using a differential equation to model the spread of a rumor.

(b) Assume your proportionality constant in the model is 1. Solve the resulting differential equation if the population we are interested in consists of 350,000 people under the following scenarios;

• nobody in the population knows the rumor

• 25,000 initially people know the rumor

• everybody initially knows the rumor Explain the behavior of the curves.

(c) If the rumor is spread by word of mouth, the rate of acquisition is proportional to the product of those who know the rumor and those who don’t. Why does this make sense?

(d) Write a differential equation that models the number of people who have acquired the rumor.

(e) Solve the new problem numerically using the initial conditions that

• 1 person knows the rumor

• that 25,000 people know the rumor

Plot your solutions over time and explain the behavior of the graphs.