A rumor spreads through a school. 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. That is, assume y' (t) = ky(1-y) where k is an unknown constant. The school has 1000 students in total. At 8 a. m. , 97 students have heard the rumor, and by noon, half the school has heard it. Using the logistic model, determine how much time passes before 90% of the students will have heard the rumor. (Note: Since y(t) denotes a proportion, 0≤ y(t) ≤ 1 for all t, and y = 1 means 100% of the students know) 90% of the students have heard the rumor after about hours. ​