Minimizing surface area...one step further

This is taking the classic 'minimizing the surface area of a cylinder' problem one step further. In the first part of the problem, it was given that the volume is 1000mL. I found the height and radius that would minimize the surface area.

The second part of the problem is this:

The material for the cans is cut from sheets of metal. The cylindrical sides are

formed by bending rectangles; these rectangles are cut from the sheet with little or no waste. But if

the top and bottom discs are cut from squares of side 2r, this leaves considerable waste metal, which

may be recycled but has little or no value to the can makers. If this is the case, show that the total

amount of metal used (including the waste metal) is minimized when h/r is 8/(pi)

The task is to show that the ratio of height to radius is 8:(pi).

How is this problem done?

Archie Meade is THE best :)

Archie Meade!!! (Clapping)(Rofl)(Clapping)(Rofl)(Clapping)

Thank you SO much! You right SUCH A LIFE-SAVER!

I actually went further than what I wrote earlier, and I knew that the apothem was the radius. I apparently can't do simple arthimetic / differeniate ><

I redid the problem myself just now (using your work as confirmation), and I ACTUALLY got the right answer this time. I've been struggling (for no good reason either) with this problem for days, literally. And now, it's SO crystal clear! Thank you SO, SO, SO much!!! This is SUCH a relief!

I wish I could bring you to everywhere I am when I have to do Math! Haha.

Could I ask you a question if you don't mind? Are you a student or professor/teacher or just a person who really loves math? I'm just really curious. All of your steps are so clear and wonderful.