Spread of a Rumor A rumor spreads through a small town. Let be the fraction of the population that has heard the rumor at time and assume that the rate at which the rumor spreads is proportional to the product of the fraction of the population that has heard the rumor and the fraction that has not yet heard the rumor. Write the differential equation satisfied by in terms of a proportionality factor Find (in units of ), assuming that 10% of the population knows the rumor at and 40% knows it at days. Using the assumptions of part (b), determine when 75% of the population will know the rumor.