## 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 th

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 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.)

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2 weeks 2021-09-13T01:43:13+00:00 1 Answer 0

Differential equation Solution Value of constant k=0.327 days^(-1)

The rumor reaches 80% at 8.48 days.

Step-by-step explanation:

We know

y(t): proportion of people that heard the rumor

y'(t)=ky(1-y), rate of spread of the rumor

Differential equation Solving the differential equation Initial conditions: Value of constant k=0.327 days^(-1)

At what time the rumor reaches 80%? The rumor reaches 80% at 8.48 days.