The rate of change (dP)/(dt) of the number of people at a carnival is modeled by the following differential equation: (dP)/(dt)=(515)/(8584)P(1-(P)/( 824)) At t=0, the number of people at the carnival is 232 and is increasing at a rate of 10 people per hour. What is the limiting value for the total number of people at the carnival as time increases? Answer:

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The rate of change  (dP)/(dt) of the number of people at a carnival is modeled by the following differential equation:  (dP)/(dt)=(515)/(8584)P(1-(P)/( 824)) At  t=0, the number of people at the carnival is 232 and is increasing at a rate of 10 people per hour. What is the limiting value for the total number of people at the carnival as time increases? Answer:

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Logistic Growth Model: The differential equation given is a logistic growth model, which is commonly used to describe populations that have a maximum carrying capacity. The equation is:  $$\frac{dP}{dt} = \frac{515}{8584}P\left(1 - \frac{P}{824}\right)$$  The limiting value, also known as the carrying capacity, is the value of $P$ at which the growth rate $\frac{dP}{dt}$ becomes zero. This happens when the term inside the parentheses equals zero:  $$1 - \frac{P}{824} = 0$$  Let's solve for $P$.Limiting Value Calculation: Add $\frac{P}{824}$ to both sides of the equation:  $$1 = \frac{P}{824}$$  Now, multiply both sides by 824 to solve for $P$:  $$824 = P$$Carrying Capacity Explanation: The value of $P$ we found is the carrying capacity of the carnival, which is the limiting value for the total number of people as time increases. This is because, in the logistic growth model, the population (or number of people in this case) approaches this value as time goes on, and the rate of increase slows down and eventually stops when the population reaches this carrying capacity.

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