There's a lot of data given in the problem, but conceptually, it's not any different than any other optimization problem. Let r be the diameter of the two disks and h the height of the cylinder.

I'm not sure what is happening in the middle paragraph - you take "-1^3" inches from the disk and "J" inches from the cylinder, and then "r" inches from the height. These should probably be numbers.

Anyway, you calculate the volume as times the radius (minus the "-1^3" lost to pressing/sealing) squared times the height (minus 2 times the "J" and the "r" lost on the top and bottom). Then you would set the volume to 57.20 gallons (which will need to be converted to cubic feet), and solve for h.

And you calculate the construction cost with the unit costs you specified (as a function of r). Then take the derivative with respect to r and set it to zero, and solve for r.

Post again in this thread if you're still having trouble.