A rumor spreads through a small town. Let y ( t ) be the fraction of the population that has heard the rumor at time t and assume that the rate at which the rumor spreads is proportional to the product of the fraction y of the population that has heard the rumor and the fraction 1 − y that has not yet heard the rumor. Write the differential equation satisfied by y in terms of a proportionality factor k . (Express numbers in exact form. Use symbolic notation and fractions where needed.) y ′ ( t ) = Find k (in units of day − 1 ), assuming that 20 % of the population knows the rumor at t = 0 and 40 % knows it at t = 3 days. (Express numbers in exact form. Use symbolic notation and fractions where needed.) k = days − 1 Using the obtained assumptions, determine when 80 % of the population will know the rumor. (Use decimal notation. Give your answer to two decimal places.)