Let :1: = m he the total biomass in millions of tonnes of the population of a. certain species of sh in the Atlantic ocean at time t. Assume that flit} is given by.r the formula 1: = he] (1 %) Mt}, where k is the rate at which the population would grew if there was an unlimited supply of [not],I L is the natural limiting size of the population is. the camping capacity of the shes' habitat, and hit} is the rate of harvesting in millions of tonnes per year at time it. (a) Suppose the fishing industry earns p dollars per million tonnes of fish harvested, and the discount rate on fish harvesting is 6. Explain why the total present value of the fishery is T = e-oth(t)pdt. (Throughout this problem, you may assume that this integral converges.) (b) Show that that I can be rewritten as T = pr(0) + pest (ka (1 - 1) - 5x) at. (c) Find the constant population r that maximizes the present value of the fishery. (d) Interpret your answer in part (c) in the case 6 > k