Calculus

# Differential Equations - Modeling

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?

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 $$t=3?$$

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 $$P(3)?$$

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 $$75$$%?

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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