Optimal design of a steel drum?

Hey guys. I have a problem due tomorrow for calculus class that is very difficult for me. To be honest I have no idea where to begin. The only thing I know is that the relevant information comes near the end of the problem. Any help would be greatly appreciated.

A 55-gallon Tight Head Steel Drum is constructed by attaching 18 gage (i.e. .0428 inches thick) steel disks to the top and bottom of a cylinder created by rolling up a 20 gage (i.e. .0324 inches thick) steel sheet.

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The vertical seam on the cylinder is welded together and the top and bottom are attached by a pressing/sealing machine. The pressing/sealing process requires approximately -1^3 i•n ches from the cylinder and J inches from the disk to be curled together and hence these inches are lost in the final dimensions. In addition, the top and bottom are set down r inches into the cylinder. For clarification, refer to the American National Standard (ANSI) specification diagram below.

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Steel can be purchased in coils (rolls) of any specified width. Construction costs can besummarized as follows:

18 gage steel is 45 cents/square foot

20 gage steel is 34 cents/square foot

welding and pressing/sealing cost 10 cents/foot

cutting steel costs 2 cents/foot.

Is the ANSI specified drum the most efficient use of material in order to obtain the required 57.20 gallon minimum volume capacity of a 55 gallon drum? Fully justify your answer.

Re: Optimal design of a steel drum?

So I am another student having serious problems with the problem. My partners and I have been working on it for a project for a while and are making little headway. We found this post but are having trouble simplifying the equation to solve for h in terms of r. Any help would be much appreciated