To test a theory of fruit flies reaching a carrying capacity, the Laurel Springs Science Club creates an environment that will attract fruit flies to a 2-cup glass jar in which is placed food and a stimulus for the fruit flies to lay eggs. The students track the population growth of the fruit flies and find the following equation 230 to fit the growth: P(t)- 1+56.5e a. Determine the initial population. -0.371 b. Sketch a graph of P(t) and use it to approximate the "carrying capacity". Explain your answer. ("Carrying capacity" is defined to be the number of organisms that a particular habitat can sustainably support due to factors such as food, environment, and other necessities available in the environment). c. What is the population after 5 days? d. How long does it take for the population to reach 180? e. How long does it take for the population to reach one-half of the carrying capacity?