After analyzing the population data of a bacteria colony, a microbiologist determines that the population of the bacterial colony can be modeled by the equation P(t) = 500e0.1t. (a) What is the initial population of the bacteria colony? (b) What function describes the instantaneous rate of change in the bacteria population after t hours? (c) What is the instantaneous rate of change in the population after 1 hour? What is the instantaneous rate of change after 8 hours? (d) How do your answers for part (c) help you to make a prediction about how long the bacteria colony will take to double in size? Make a prediction for the number of hours the population will take to double, using your answers for part c, and/or other information. Give justifications for your prediction. (e) Determine the actual doubling time (the time that the colony takes to grow to twice its initial population). (f) Compare your prediction for the doubling time with the calculated value. If your prediction was not close to the actual value, what factors do you think might account for the difference? (g) When is the instantaneous rate of change equal to 500 bacteria per hour?