Somewhere in Chiwempala district there is a steel plant located next to a river. This plant produces and exports steel bars (B). It also produces sludge (S) as a by-product. The cost function (? ) of steel bar production is ? = ? ?+ ?5 - 5 2 + 35 , and the price for the bar of steel is P. = 169 -2? . Farther downstream is a traditional fishing area where fishermen catch fish (F) for sale to local restaurants with cost function of ? = 5F 2 + 2FS + 35 2 where F is fish and S is sludge dumped into the river that the fishermen have to remove for the fishing to remain profitable. The sludge is due to the carelessness of the steel bar smelting plant. The price of fish is given as P = 226 . The two enterprises operate as separate entities without caring what each one is doing. Under the given circumstances, derive the: equations for optimal output conditions of steel plant and fishing enterprises under no cooperation [15 marks] ii) levels of output for steel bars, sludge and fish [15 marks] iii) levels of profit for steel plant and fishing enterprises [10 marks]