Amongst all the cilinders inscribed in a sphere of radius 1, I have to find that one of maximal surface.
Every cilinder is characterized by a value for $\alpha$, the angle between the the vertical line for the center of the sphere and a line that goes through the edge of the base of the cilinder.
I've found that : $$S(\alpha)=2*A_{BASE}+2 \pi*\sin \alpha*2*\cos \alpha=2 \pi* (\sin \alpha)^2+4 \pi*\sin \alpha*\cos \alpha$$ $$S'(\alpha)=4 \pi*(\sin \alpha *\cos\alpha+(\cos \alpha)^2-(\sin \alpha)^2)=0$$ I've found that $\alpha =\operatorname{arctg}(-2)/2$ and $S \approx 10,09 $
My book suggests $\pi *(1+ \sqrt{5})$ for this maximum value of surface and $10,09 \approx \pi *(1+ \sqrt{5})$. But I don't know how to reach the value exactly. Perhaps I'm ignoring something.