Please note that the formula for the surface area can be rewritten as A = pi \times \sqrt{r^2 \times ( r^2 + h^2 )} (LaTeX notation). To maximise A it suffices to maximise the radicand. Since it's quite easy to substitute r^2 by using the equation for the cone volume V, we have a function of h rather than a function of r. It's derivative can be calculated essentially by using the product, quotient and addition rules for derivatives. The numerator of the derivative is given by 3V \times ( h^3 \pi - 6V ). From this we have a zero derivative if h^3 \pi = 6V. Again use the equation for the cone volume to find that h^2 = 2 r^2 or h/r = \sqrt{2}.
Please note that the formula for the surface area can be rewritten as A = pi \times \sqrt{r^2 \times ( r^2 + h^2 )} (LaTeX notation). To maximise A it suffices to maximise the radicand. Since it's quite easy to substitute r^2 by using the equation for the cone volume V, we have a function of h rather than a function of r. It's derivative can be calculated essentially by using the product, quotient and addition rules for derivatives. The numerator of the derivative is given by 3V \times ( h^3 \pi - 6V ). From this we have a zero derivative if h^3 \pi = 6V. Again use the equation for the cone volume to find that h^2 = 2 r^2 or h/r = \sqrt{2}.