Math 1261: Calculus I Brown Optimization Problems 1. Find a positive number such that the sum of it and its reciprocal is as small as possible. 2. A box with a square base and open top must have a volume of 32,000 cm3 . Find the dimensions of the box that minimize the amount of material used. 3. A piece of wire 3 cm in length is cut into two pieces. One piece is bent into the shape of a circle and the other is bent into the shape of a square. Determine how to obtain the maximum area enclosed by the two figures. 4. Find the area of the largest rectangle that can be inscribed in a right triangle with legs of length 3 cm and 4 cm if two sides of the rectangle lie along the legs. 5. A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible 1 volume of such a cylinder. (Vcone = πr2 h and Vcylinder = πr2 h) 3 6. A woman at a point A on the shore of a circular lake with radius 2 miles wants to arrive at a point C diametrically opposite A on the other side of the lake in the shortest possible time. She can walk at a rate of 4 miles per hour and row a boat at a rate of 2 miles per hour. How should she proceed? 7. For the given cost and demand functions, find the production level that will maximize profit. C(x) = 680 + 4x + 0.01x2 , p(x) = 12 − x 500 8. Find the production level at which the marginal cost function starts to increase. C(x) = 0.0002x3 − 0.25x2 + 4x + 1500 9. A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at $ 55, the average attendance had been 27,000. When ticket prices were lowered to $ 8, the average attendance rose to 33,000. Find the demand function, assuming it is linear, and determine how much the team should charge to maximize revenue. Note: Some useful relationships for economic and business application. P , C, R, and p denote profit, cost, revenue, and demand functions. P (x) = R(x) − C(x), Profit=Revenue-Cost R(x) = xp(x), Revenue=(units sold)(price per unit) c(x) = C(x) , x Average Cost = (Cost per unit)/(units sold)