The rate of change (dP)/(dt) of the number of people in a park is modeled by the following differential equation: (dP)/(dt)=(4085)/(36816)P(1-(P)/( 860)) At t=0, the number of people in the park is 236 and is increasing at a rate of 19 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 in a park is modeled by the following differential equation:  (dP)/(dt)=(4085)/(36816)P(1-(P)/( 860)) At  t=0, the number of people in the park is 236 and is increasing at a rate of 19 people per hour. Find  lim_(t rarr oo)P^(')(t). Answer:

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Analyze Differential Equation: Analyze the given differential equation. The differential equation given is (dP)/(dt) = (4085)/(36816)P(1 - (P)/(860)). This is a logistic growth model where the growth rate of the population P is proportional to both the current population and the difference between the current population and the carrying capacity, which in this case is 860.Determine Steady State: Determine the steady state of the population. In the logistic growth model, the steady state, or equilibrium, occurs when the rate of change of the population (dP)/(dt) is zero. This happens when P is at its carrying capacity or when P is zero. Since we are looking for the limit as t approaches infinity, we are interested in the carrying capacity, which is P = 860.Calculate Limit: Calculate the limit of P'(t) as t approaches infinity. As t approaches infinity, the population P approaches its carrying capacity. Therefore, the term (1 - (P)/(860)) in the differential equation will approach zero, because P will be very close to 860. This means that the rate of change of the population (dP)/(dt) will also approach zero.Conclude Result: Conclude the limit of P'(t) as t approaches infinity. Since the rate of change of the population (dP)/(dt) approaches zero as t approaches infinity, we can conclude that lim_(t rarr oo)P'(t) = 0.

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