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

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    2021-09-13T01:44:15+00:00

    Answer:

    Differential equation

    \frac{dy}{dt} =ky(1-y)

    Solution

    y=\frac{1}{1+4e^{-0.327t}}

    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

    \frac{dy}{dt} =ky(1-y)

    Solving the differential equation

    \frac{dy}{y(1-y)}=k\cdot dt \\\\\int \frac{dx}{y(1-y)} =k \int dt \\\\-ln(1-\frac{1}{y} )+C_0=kt\\\\1-\frac{1}{y} =Ce^{-kt}\\\\\frac{1}{y} =1-Ce^{-kt}\\\\y=\frac{1}{1-Ce^{-kt}}

    Initial conditions:

    y(0)=0.2\\y(3)=0.4\\\\y(0)=0.2=\frac{1}{1-Ce^0}\\\\1-C=1/0.2\\\\C=1-1/0.2= -4\\\\\\y(3)=0.4=\frac{1}{1+4e^{-3k}} \\\\1+4e^{-3k}=1/0.4\\\\e^{-3k}=(2.5-1)/4=0.375\\\\k=ln(0.375)/(-3)=0.327\\\\\\y=\frac{1}{1+4e^{-0.327t}}

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

    At what time the rumor reaches 80%?

    y(t)=0.8=\frac{1}{1+4e^{-0.327t}} \\\\1+4e^{-0.327t}=1/0.8=1.25\\\\e^{-0.327t}=(1.25-1)/4=0.0625\\\\t=ln(0.0625)/(-0.327)=8.48

    The rumor reaches 80% at 8.48 days.

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