First Order Differential Equations

Differential Equations - Modeling


Let P(t)P(t) represent the amount of chemical a factory produces as a function of time tt (in hours). The rate of change of chemical production satisfies the differential equation P(t)=ln3×P(t)(1P(t)3).P'(t) = -\ln 3 \times P(t) \left( 1 - \frac{P(t)}{3}\right). If the factory alarm is raised when chemical production exceeds 44 in 44 hours, which of the following inequalities represents the maximum initial amount P(0)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)P(t) at time t.t. If P(t)P(t) satisfies the differential equation P(t)=ln2×P(t)(1P(t)12)P'(t) = \ln 2 \times P(t) \left( 1 - \frac{P(t)}{12}\right) and the initial number of cells is P(0)=9,P(0)=9, what is the approximation for the number of cells in the culture at time t=3?t=3?

Suppose the ratio of healthy cells to infected cells in a petri dish at time tt is represented by P(t)P(t). If P(t)P(t) satisfies the logistic differential equation P(t)=ln5×P(t)(1P(t)7)P'(t) = -\ln 5 \times P(t) \left( 1 - \frac{P(t)}{7}\right) and P(0)=2P(0)= 2, what is the value of P(3)?P(3)?

Suppose the percentage of people surviving a dangerous virus at time tt is approximated by P(t)P(t), where P(0)=100.P(0)=100. If P(t)P(t) satisfies the logistic differential equation P(t)=6×P(t)(1P(t)20),P'(t) = 6 \times P(t) \left( 1 - \frac{P(t)}{20}\right), at what value of tt is the survival rate 7575%?

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


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