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MCV4U Optimization Problems in Economics and Science Profit, cost and revenue are quantities whose rates of change are measured in terms of the number of units produced or sold. Economic situations usually involve minimizing costs or maximizing profits. Recall: Cost Function C(x) = (number of items)*(cost per item) Revenue Function R(x) = (number of items)*(price per item) Profit Function P(x) = R(x) - C(x) Where x is the number of items. Typically, when price increases, the number of items sold decreases. The variable typically used is the "number of price increases". The break-even point is the change in profit from a negative to positive or vice-versa. So at the break-even point, P(x) = 0 or R(x) = C(x). Marginal profit, marginal revenue and marginal cost are the rates of change of profit, revenue, and cost, from selling or producing one more unit. Eg. 1: A rectangular piece of land is to be fenced using two kinds of fencing. Two opposite sides will be fenced using standard fencing with a cost of $6/m, while the other 2 require heavy duty fencing that cost $9/m. What are the dimensions of the rectangular lot of greatest area that can be fenced for a cost of $9000.Eg.2: A railroad between two cities carries 10 000 passengers per year when the fare is $50. If the fare goes up, the number of passengers will decrease, since more people will drive. It is estimated that each $10 increase in fare will result in 1000 fewer passengers per year. The train can not carry more than 13000 passengers per year. a) At what fare will the train have no passengers? b) What fare will maximize revenue? When revenue is maximized, how many people will ride the train each year?Optimization Problems in Economics and Science Eg.1: A cylindrical chemical storage tank with a capacity of 1000 m' is going to be constructed in a warehouse that is 12 m by 15 m by 11 m in height. The specifications call for the base to be made of sheet steel that costs $100/m , the top to be made of sheet steel that costs $50/m and the wall to be made of sheet steel that costs $80/m. a) Determine whether it is possible for a tank of this capacity to fit in the warehouse. If it is possible, state the restrictions on the radius of the tank. b) If fitting the tank in the warehouse is possible, determine the proportions that meet the conditions and that minimize the cost of the steel for construction