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 to fit the growth: P(t)= (230)/(1+56.5e^(-0.37t))

a) Determine the initial population.

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?