The formulas for the surface area and volume of an equal right cone are:
Area = pi*r^2*(sqrt(5)+1)
Volume = 2/3*pi*r^3
We want the difference to be a maximum:
A-V = pi*r^2*(sqrt(5)+1)-2/3*pi*r^3
Let’s differentiate:
pi*2r*(sqrt(5)+1)-2/3*pi*3r^2
2*pi*r*(sqrt(5)+1-r)
The (Area-Volume) will be at a maximum when the above expression is equal to 0, or more specifically, when the term in brackets is equal to 0, which is obviously when r = sqrt(5)+1.
To answer the original question, the surface area will be pi*(sqrt(5)+1)^3 ~ 106.464.