DESIGNING A DUMPSTER Part I For this project you have to locate a trash dumpster in order to study its shape and construction. Then you attempt to determine the dimensions of a container of similar design that minimize construction cost. 1. First locate a trash dumpster in your area. Carefully study and describe all details of its construction, and determine its volume. a. Include a sketch of the container with the dimensions marked on it (it doesn't have to be in scale). b. Include a picture of the dumpster. 2. While maintaining the general shape and method of construction, determine the dimensions such a container of the same volume should have in order to minimize the cost of construction. Use the following assumptions in your analysis: a. The sides, back, and front are to be made from 12-gauge (0.1046 inch thick) steel sheets, which cost $0.80 per square foot (including any required cuts or bends). b. The base is to be made from a 10-gauge (0.1345 inch thick) steel sheet, which costs $0.95 per square foot. c. The lid costs $70.00 each, regardless of dimensions. d. Welding costs approximately $0.25 per foot for material and labor combined. Give justification of any further assumptions or simplifications made of the details of construction. 3. Describe how any of your assumptions or simplifications may affect the final result. 4. If you were hired as a consultant on this investigation, what would your conclusion be? Would you recommend altering the design of the dumpster? If so, describe the savings that would result. Part II Let's assume you have received an order of a dumpster which needs to have a rectangular base and a volume of 200 cubic feet. Label the dumpster's width in feet by x and its depth in feet (from front to back) by y. Its height at the front is z feet, while it is required to be one foot higher at the back to let the rain water flow off the lid, see figure below. Use the same information for the material required and the cost from Part I to find the dimensions of the dumpster that will minimize the cost. Hints: Express the cost required for building a dumpster as a function C(x, y, z). Use the fact that the volume of the dumpster must be constant to eliminate the variable z from the expression for C(x, y, z). Find the resulting function of two variables and call it c(x, y). Find the critical points