The rate of change (dP)/(dt) of the number of students who heard a rumor is modeled by the following differential equation: (dP)/(dt)=(30804)/(90965)P(1-(P)/( 906)) At t=0, the number of students who heard the rumor is 115 and is increasing at a rate of 34 students per hour. Find lim_(t rarr oo)P^(')(t). Answer:

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The rate of change  (dP)/(dt) of the number of students who heard a rumor is modeled by the following differential equation:  (dP)/(dt)=(30804)/(90965)P(1-(P)/( 906)) At  t=0, the number of students who heard the rumor is 115 and is increasing at a rate of 34 students per hour. Find  lim_(t rarr oo)P^(')(t). Answer:

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Identify Differential Equation: The given differential equation is (dP)/(dt) = (30804)/(90965)P(1 - (P)/(906)). This is a logistic growth model where the growth rate of P is proportional to both the current value of P and the difference between P and the carrying capacity, which in this case is 906. To find the limit of P'(t) as t approaches infinity, we need to understand the behavior of the logistic function as time goes on.Understand Logistic Growth Model: In a logistic model, as t approaches infinity, the value of P(t) approaches the carrying capacity. This is because the term (1 - (P)/(906)) will approach zero as P approaches 906, causing the rate of change P'(t) to approach zero as well.Find Limit of Growth Rate: Therefore, the limit of P'(t) as t approaches infinity is 0, because the population P will stabilize at the carrying capacity, and there will be no further change in the number of students who have heard the rumor.

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