$5.(10$ marks$)$ A rumor spreads through a school. Let $\backslash ($ y$($t$)\backslash )$ be the fraction of the population that has heard the rumor at time $\backslash ($ t $\backslash )$ and assume that the rate at which the rumor spreads is proportional to the product of the fraction $\backslash ($ y $\backslash )$ of the population that has heard the rumor and the fraction $\backslash (1-$y $\backslash )$ that has not yet heard the rumor.

The school has $1000$ students in total. At $8$ a$.$m$.,96$ students have heard the rumor, and by noon, half the school has heard it$.$ Using the logistic model explained above, determine how much time passes before $\backslash (90\backslash \%\backslash )$ of the students will have heard the rumor.