Question

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Optimization problems 1. (5 points) Find two nonnegative numbers whose sum is 25 and so that the product of one number and the square of the other number is a maximum. 2. (5 points) Build a rectangular pen with two parallel partitions using 300 feet of fencing. What dimensions will maximize the total area of the pen? (5 points) An open rectangular box with square base is to be made from 48 ft.2 of material. What dimensions will result in a box with the largest possible volume? 3. (5 points) An open rectangular box with square base is to have a volume of 48 ft.3. What dimensions will result in a box with the least possible surface area? 4. (10 points) A container in the shape of a right circular cylinder with no top has surface area 6 r ft.2 What height h and base radius r will maximize the volume of the cylinder? 5. (15 points) A sheet of cardboard 4 ft. by 5 ft. will be made into a box by cutting equal- sized squares from each corner and folding up the four edges. What will be the dimensions of the box with largest volume? 6. G (10 points) Find the point (x, v) on the graph of y nearest the point (6,0. (10 points) A landscape architect plans to enclose a rectangular region with an area of 3000 square feet against a long, straight wall. She will use cacti that cost $40 per foot on the two sides that touch the wall and shrubs costing $30 per foot on the fourth side opposite the wall. Find the dimensions of the region that will cost the least to enclose. 9. (15 points) A cylindrical can is to hold 20r m.3 The material for the top and bottom costs $10/m.2 and material for the side costs $8/m.2 Find the radius r and height h of the most economical can. 10. (15 points) Construct a window in the shape of a semi-circle over a rectangle. If the distance around the outside of the window is 12 feet, what dimensions will result in the rectangle having largest possible area?

Answer 1