Minimize the surface area A of a tin can of fixed volume V.

Area of the tin can: A = 2pr² + 2prh

Volume of the tin can: V = pr²h

We introduce a scale-invariant variable (dimensionless quantity):

We express h and r as function of V and t:

We express A as a function of V and t:

V is fixed and therefore min A is attained for C(t) minimum, that is for t^-2/3 + t^1/3 minimum.

Plotting of:

y = t^-2/3 + t^1/3

Minimum of f(t) = t^-2/3 + t^1/3

The minimum of f(t) is obtained for df(t) / dt = 0

The surface area A of a tin can of fixed volume is minimized when  h = 2r.