I have worked out how to find the optimum surface area dimensions for a full cone with a set volume, but am unsure how to go about this when the cone is truncated (bottom pointed end cut off).




I've derived the below volume and surface area formulas for a frustum cone.


V = ⅓.Pi.h(R^2 + Rr + r^2)


SA = Pi(R + r)[(R-r)^2 + h^2]^½


I then found what h = using the first volume formula:


h = (3V)/Pi(R^2 + Rr + r^2)


Then I subbed h into the SA formula such that:


SA = Pi(R + r)[(R-r)^2 + (3V/Pi(R^2 + Rr + r^2))]^½


I think that I am supposed to use differential calculus now to find the critical values/local minimum but am not sure how to do this because there are two different radii values?