The rate of change, \(\dfrac{dP}{dt}\), of the number of people on an ocean beach is modeled by a logistic differential equation. The maximum number of people allowed on the beach is 1200. At 10 A.M., the number of people on the beach is 200 and is increasing at the rate of 400 people per hour. Which of the following differential equations describes this situation?

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A logistic differential equation has the general form:

$$ \frac{dP}{dt}=kP\left(1-\frac{P}{L}\right) $$

The carrying capacity, \(L\), is \(1200\).

$$ \frac{dP}{dt}=kP\left(1-\frac{P}{1200}\right) $$ $$\frac{dP}{dt} = kP\left(\frac{1200}{1200}-\frac{P}{1200}\right) $$ $$ \frac{dP}{dt}= \frac{kP}{1200}(1200-P) $$

We know that when \(P=200\), \(\dfrac{dP}{dt}=400\).

$$ 400 = \frac{k(200)}{1200}(1200-200) $$ $$ k = \frac{12}{5} $$ $$ \frac{dP}{dt} = \frac{(\frac{12}{5})P}{1200}(1200-P) $$ $$ \boxed{\dfrac{dP}{dt}=\dfrac{1}{500}P(1200-P) } $$