# Height and Radius for maximum volume cylinder of given surface area

I've got a school project, in which I'm supposed to get a radius of a base and a height of a cylinder while having the maximal volume possible. Surface area of the cylinder is known.

I found a great solution on this forum (https://math.stackexchange.com/a/1449593/621973), but the problem is that I need to explain my teacher, how I turned $$\frac k{(k+1)^{3/2}}$$ from V(k) into $$f(k)=\ln\frac{k}{(k+1)^{3/2}}$$ By the way, I can't give her another solution so I really need to find explanation for this one.

I'll be glad for every idea.

P.S.: Sorry for my english.

You can maximize $$f(k)=\frac k{(k+1)^{3/2}}$$ or its logarithm $$g(k)=\log \left(\frac{k}{(k+1)^{3/2}}\right)=\log(k)-\frac 32 \log(k+1)$$ $$g(k)$$ is a better choice since its derivative is easier to get.