Final answer:
The rate of change of the bat population at t = 0 cannot be determined without specific information, but models can be used if data were available. The estimated size of the lemur population can be calculated through "mark and recapture" techniques, and cell populations can be estimated based on doubling times.
Step-by-step explanation:
Without additional information about the bat population's specific growth rate or a mathematical model describing its dynamics over time, we cannot determine the exact rate of change of the bat population at t = 0. However, if we had a formula similar to the population growth model for humans indicated in the given information, we could apply the same principles to the bats, assuming that the rate of growth or decline is given by a constant percentage. For example, if the bat population were decreasing by a certain percentage annually due to the effects of a pesticide on their food source, we could use a logarithmic growth model to calculate the rate of change at any given time.
In the mark and recapture scenario for the lemurs, an estimated population size can be calculated using the formula N = (number marked first capture × total second capture) / (number marked second capture), which in this case yields N = (37 × 49) / 11. This estimation provides a rounded number for the size of the lemur population.
In the scenario where a population of cells has a doubling time of 30 minutes, after 2 hours, which is equivalent to 4 doubling periods, the population would quadruple in size calculated by 24 = 16 times the original population, resulting in 1.6 × 106 cells assuming no cell death.