How to optimise a toothpaste tube / moisturiser squeeze tube? I am trying to find the least surface area to volume ratio. [closed]

So that the least amount of material is used/wasted in making a tube. I initially tried modelling it using area of revolution but it's impossible since the tube is tapered at the end.

I then decided to split the tube into two parts so I found an equation for the top part of the tube.

v=(4+pi)/12)pi x radius squared x height

Surface area = 2 pi r h

Does anyone know how I can optimise these two equations using Lagrange Multiplier or just in general

or other methods I can use to optimise the tube?

thank you

The problem as stated is not well defined. The surface area to volume ratio of a closed solid typically decreases with increasing volume, implying you should let volume go to infinity to minimize the ratio. For instance, in the case of a sphere, the ratio is $\frac{4\pi r^2}{\frac{4}{3}\pi r^3}\propto \frac{1}{r}$.

A more meaningful question would be what closed shape minimizes surface area subject to having some given volume. The answer here would then be a sphere (a consequence of the isoperimetric inequality).

See here and here if interested.