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Problem 11 Medium Difficulty
One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction $ y $ of the population who have heard the rumor and the fraction who have not heard the rumor.
(a) Write a differential equation that is satisfied by $ y. $
(b) Solve the differential equation.
(c) A small town has 1000 inhabitants. At $ 8 AM, $ 80 people have heard a rumor. By noon half the town has heard it. At what time will $ 90% $ of the population have heard the rumor?
Answer
a) (a) Our assumption is that $\frac{d y}{d t}=k y(1-y),$ where $y$ is the fraction of the population that has heard the rumor.
b) $$y=\frac{y_{0}}{y_{0}+\left(1-y_{0}\right) e^{-k t}}$$
c) $90 \%$ of the population will have heard the rumor by 3: 36 PM.
Topics
Differential Equations
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Video Transcript
Hey, it's Claire. So when you right here. So we're gonna let the number of people who have heard the rumors A and those who haven't be so we have why is equal to a over a plus b So are finding the rate of the spread. So it's d y over a dp, and we're seeing that the fraction of the population who have not heard the rumor is gonna be be over a plus B. And we get this to be a plus B minus day over a plus B. This is equal to one minus a over a plus B when we make this be equal to why we get one minus Why? So we get this to equal to K terms. Why comes one minus? Why, when Kate iss proportionally constant for part B, we have our equation from that we got from party. So we're gonna make people to why? And it's equal to one. So the solution is why of tea is equal to one over one plus que e to the negative. Katie, where a is equal to one minus y subzero over. Why subzero? So this is equal to one over one plus one minus y subzero Over. Why subzero comes e to the negative ke t which is equal to y subzero allover by sub zero plus one minus Why subzero eats a negative. Katie for part c, we're gonna make t is equal to zero. Got a t m. So we have Y sub zero is equal to 80 over 1000 which is 0.8 And by noon, we know the half of the town has heard the rumor. So you get why a four is equal to 0.5. We're getting our equation from part B, and we're just plugging it in using 0.0 a when. Why of zero when we got 0.0 Kate all over 0.8 plus 0.92 times e to the negative. Katie, we have zero point frying as equal to 0.8 all over 0.0 K plus 0.92 eats and negative for K. So the end of getting K is equal to negative 2.442 over negative for, and we get 0.612 Oh, and then we have why of tea is equal to 0.0 a all over 0.0. A school of 0.92 eats the negative 0.612 tea, and then we end up getting a T value of about 7.6, which is around seven hours and 36 minutes. So 90% of the rumor Hospira about 3:36 p.m.
Topics
Differential Equations
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