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.

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