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But differentiation is not really Stage 4 mathematics, though very close, so we don't use it in solutions because that would be unfair to students who haven't seen the idea before, it would be like suddenly changing into a language you haven't had a chance to learn yet.
If that's got you interested why not take a look at Vicky Neale's article : Introduction to Differentiationthe height, which will be determined by our choice of radius, so that we get the chosen target volume
the slant length, which we'll find using r and h and Pythagoras,
and the surface area, for which there's a great little formula.
Incidentally, if you don't know where that surface area formula comes from it may be good to take a moment to look at that. Flatten the curved surface out to get a sector (how do you know it's a sector ?). The radius will be the cone's slant length, so you can calculate the area of the whole circle. To know the proportion that the sector is of that circle compare the sector arc, which is the cone's base circumference, with the circumference of this new circle, radius s.
Can you see what each column does ? Click on a cell and check the formula.