Glass optimization

Calculus Level pending

You wish to construct a glass in the shape of a perfect cylinder so that it will hold the maximum amount of liquid possible. You have a sheet of plastic with the surface area of 40\(\pi \) which you plan to reshape into a fully functional drinking glass. The thickness of the material does not change in the process. Given this data, what is the ratio between the height and the radius of the finished product, i.e. \(\frac{ h }{ r } \)?

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