A boutique beer brewery produces 2 types of beers, Darkale and Lightale daily with a total cost function: TC = SQD + (2;. x QL + 20]: where: Q1; is the quantity of the Darkale beer [in kegs] and Q; is the quantity of the Lightale beer {in kegs]. The prices that cab be charged are determined by supply and demand forces and are inuenced by the quantities of each type of beer according to the following equations: PD = 13 30;. + Q; forthe price [in dollars per keg] ofthe Darkale beer and P; = 1? + 40;. Q; for the price [in dollars per keg] ofthe Light-ale beer. The total revenue is given by the equation: TR = PD x (2;, + PL x Q]; and the profit given by the equation Profit = TR TC {a} Use a substitution ofthe prioe variables to express the profit in terms of Q3 and (2; only [2 marks] {b} Using the method of Lagrange Multipliers nd the maximum profit when total production [quantity] is restricted to 113 kegs. Note QB or QL need not be a whole number. [5 marks] {c} Estimate the new prot ifthe production restriction is increased by 1 keg and relate your findings to the value of the Lagrange multiplier. [2 marks] [Hint: Replace the production limit {113] with B a new variable.]