I am trying to help my friend. This is his problem related to constructing a dumpster, so it can minimize construction cost :
For this project we locate a trash dumpster in order to study its shape and construction. We then attempt to determine the dimensions of a container of similar design that minimize construction cost."
-
(Already located, measured, and described a dumpster found).
-
"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:
-
The sides, back, and front are to be made from 12-gauge (0.1046 inch thick) steel sheets, which cost $0.70 per square foot (including any required cuts or bends).
-
The base is to be made from a 10-gauge (0.1345 inch thick) steel sheet, which costs $0.90 per square foot.
-
Lids cost approximately $50.00 each, regardless of dimensions.
-
Welding costs approximately $0.18 per square foot for material and labor combined.
-
Give justification of any further assumptions or simplifications made of the details of construction.
-
-
Describe how any of your assumptions or simplifications may affect the actual result.
-
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."
Best Answer
You know the volume $V$ that is required. It should be clear that the footprint should be square, as that has the least perimeter for the given area. So you need to choose the side of the square, $s$. The height is then $h=\frac V{s^2}$ Use the data in 2 to write an equation for cost as a function of height. Take the derivative, set to zero, and find the optimum side of the square.