A culture of algae grows in a nutrient-rich solution. The rate of change of the population P (measured in cells) with respect to time t (in hours) is directly proportional to the size of the population, and the initial population is P(0)=500.
(a) Explain what information about the long–term behavior of the population can be inferred from the sign of the constant of proportionality k in the differential equation dtdP=kP. Support your answer with reasoning in the context of the algae culture.
(b) Derive an explicit expression, in terms of k, for the amount of time required for the population to triple from its initial size.
(c) If laboratory measurements later determine that k=0.04 hr−1, calculate the number of hours needed for the population to triple and interpret this result in the biological context.