These are equations, Calculus-style. From modeling real-world phenomenon, from the path of a rocket to the cooling of a physical object, Differential Equations are all around us.

Let \(P(t)\) represent the amount of chemical a factory produces as a function of time \(t\) (in hours). The rate of change of chemical production satisfies the differential equation \[P'(t) = -\ln 3 \times P(t) \left( 1 - \frac{P(t)}{3}\right).\] If the factory alarm is raised when chemical production exceeds \(4\) in \(4\) hours, which of the following inequalities represents the maximum initial amount \(P(0)\) of chemical that guarantees the alarm will not be raised?

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Suppose the number of cells in a culture is approximated by \(P(t)\) at time \(t.\) If \(P(t)\) satisfies the differential equation \[P'(t) = \ln 2 \times P(t) \left( 1 - \frac{P(t)}{12}\right)\] and the initial number of cells is \(P(0)=9,\) what is the approximation for the number of cells in the culture at time \(t=3?\)

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Suppose the ratio of healthy cells to infected cells in a petri dish at time \(t\) is represented by \(P(t)\). If \(P(t)\) satisfies the logistic differential equation \[P'(t) = -\ln 5 \times P(t) \left( 1 - \frac{P(t)}{7}\right)\]
and \(P(0)= 2\), what is the value of \(P(3)?\)

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Suppose the percentage of people surviving a dangerous virus at time \(t\) is approximated by \(P(t)\), where \(P(0)=100.\) If \(P(t)\) satisfies the logistic differential equation \[P'(t) = 6 \times P(t) \left( 1 - \frac{P(t)}{20}\right),\] at what value of \(t\) is the survival rate \(75\)%?

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Suppose the population in a park at time \(t\) is given by \(P(t)\), where \[P'(t) = -0.5 \times P(t) \left( 1 - \frac{P(t)}{5}\right).\] If \(P(0) = 12,\) at what time \(t\) does the population in the park reach \(16?\)

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