A biologist stocked a lake with 45 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 7,000. The number of fish tripled in the first year.
(a) Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population as a function of time.
(b) How long will it take for the population to increase to 3,500?