The rate of change (dP)/(dt) of the number of people on an island is modeled by the following differential equation: (dP)/(dt)=(3920)/(27101)P(1-(P)/( 784)) At t=0, the number of people on the island is 123 and is increasing at a rate of 15 people per hour. Find lim_(t rarr oo)P(t). Answer:

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The rate of change  (dP)/(dt) of the number of people on an island is modeled by the following differential equation:  (dP)/(dt)=(3920)/(27101)P(1-(P)/( 784)) At  t=0, the number of people on the island is 123 and is increasing at a rate of 15 people per hour. Find  lim_(t rarr oo)P(t). Answer:

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Analyze the differential equation: Analyze the given differential equation. The differential equation is given by (dP)/(dt) = (3920)/(27101)P(1 - (P)/(784)). This is a logistic growth model, where the growth rate of the population is proportional to both the current population and the room left for growth (the difference between the current population and the carrying capacity). The carrying capacity in this case is 784, which is the maximum population that the environment can sustain.Determine behavior as t approaches infinity: Determine the behavior of P(t) as t approaches infinity. For logistic growth models, as time goes on, the population P(t) approaches the carrying capacity, because the term (1 - (P)/(784)) becomes smaller as P gets closer to 784, reducing the growth rate. Eventually, the growth rate becomes zero when P reaches 784, and the population stabilizes.Calculate limit as t approaches infinity: Calculate the limit of P(t) as t approaches infinity. Since the carrying capacity is 784, the limit of P(t) as t approaches infinity is the carrying capacity itself, because the population cannot exceed this value in the logistic model.

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