Could you help me with this "Externality" problem?

Consider two oil refineries that both produce fuel, which has a market price of $3 per gallon. Assume that each refinery uses $1 in raw inputs (crude oil, electricity, labor) to produce 1 gallon of fuel. In addition, each plant produces smog, which creates $0.02 of environmental damage per cubic foot. The amount of smog per gallon of fuel produced differs at the two plants: $1 = 2yi and 42 where y1 and y2 denote the number of gallons of fuel produced at each plant, and $1 and so denote the amount of smog generated. Plant 2 pollutes much less than plant 1 for a given production. Assume each plant can produce only up to 200 gallons. (a) (10 pts) Set up the problem for each plant, and determine production, pollution (smog) and damage when there is no policy. (b) (15 pts) Set up the problem for each plant, and determine production, pollution (smog) and damage that maximizes welfare. (Hint: To solve it, imagine that each plant faces private costs plus the social costs of production). (c) (10 pts) Imagine that a Pigouvian tax (of $0.02 per unit of smog) is imposed on the plants. If you were to determine production, pollution (smog) and damage again, how does your answer compare to part (b)? Why