An umbrella-manufacturer in a perfectly competitive market (price-taker) faces an inverse demand curve given by P(Q) = 10 - Q. The cost-function of the firm is given by C(Q) = 2Q + =Q. Suppose that producing an umbrella is a very polluting process, and the marginal external damage cost of every umbrella is estimated to be 4 dollars, so that the total health cost of producing Q umbrellas is C (Q) = 4Q.Suppose the government charges the umbrella-manufacturer a tax T for every umbrella it produces, so that the cost function of the firm becomes CT(Q) = 20 + =Q2 + TQ. Find the new marginal cost function of the firm and the equilibrium quantity, Q, and price, PT, that would be observed in the market, in terms of T. What must be the value of t chosen by the government to ensure the quantity produced by the firm matches the socially optimal quantity you found in E)