A boutique beer brewery produces 2 types of beers, Darkale and Lightale daily with a total cost function: TC = 5121, + (3;. x QL + EQL where: (2;, is the quantity of the Dark-ale beer [in kegs] and Q; is the quantity of the Lightale beer {in kegs]. The prices that can 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 39o + Q; for the price [in dollars per keg] ofthe Darkale beer and PL = 1? + 4.9;. QL 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 {2L and the profit given by the equation Profit = TR TC {a} Use a substitution ofthe prioe variables to express the profit in terms of Q\" and Q only [2 marks] {b} Using the method of Lagrange Multipliers nd the maximum profit when total production [quantity] is restricted to 113 kegs. Note Q3 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.]