Show all work for the following Optimization problems. LPW - W W (1) Maximizing Area (i) Of all rectangles with a fixed perimeter of P feet, which one has the maximum area A? (Give the dimensions in terms of P.) (ii) Let P = 40 feet and give the dimensions using your solution in part (i). (iii) Sketch the area function A that was maximized in part (i). Use a reasonable domain. Label axes appropriately, including units. R- 2W A = LYW P - R. alt 2W (2) Minimizing Perimeter- (1) Of all rectangles with a fixed area of A inches squared, which one has the smallest perimeter P? (Give the dimensions in terms of A. ) (ii) Let A = 100 inches squared and give the dimensions using your solution in part (i). (iii) Sketch the perimeter function P that was minimized in part (i). Use a reasonable domain. Label axes appropriately, including units. 40 - 264 2X (3) Minimizing Surface Area (i) Of all boxes with a square base and a fixed volume V, which one has the minimum surface area As? ( Give its dimensions in terms of V.) (ii) Let V = 1000 meters cubed and give the dimensions using your solution in part (i). (iii) Sketch the surface area function As that was minimized in part (i). Use a reasonable domain. Label axes appropriately, including units. (4) Maximizing Volume (i) Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 3 ft by 4 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way. (ii) Suppose that in part (i) the original piece of cardboard is a square with sides of length l. Find the volume of the largest box that can be formed in this way (5) Minimizing Distance (i) Find the point P on the line y = 3x that is closest to the point (50,0). (ii) What is the least distance between P and (50,0)? (6) Minimizing Cost. A builder wants to construct a cylindrical barrel with a capacity of 32n fts. The cost per square foot of the material for the side of the barrel is half that of the cost per square foot for the top and bottom. Determine the dimensions of the barrel that can be constructed at a minimum cost in terms of material used