Question: 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

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.

(a) Write the differential equation satisfied by y in terms of a proportionality factor k.

(b) Find k (in units of day⁻¹), assuming that 10% of the population knows the rumor at t = 0 and 40% knows it at t = 2 days.

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