Question

A right circular cylinder is inscribed in a sphere of radius r. We want to find the dimensions of such a cylinder with the largest possible surface area (your answers may depend on r).

The dimensions of the cylinder with the largest possible surface area are?

Answer 1

\[4x^2+y^2=(2r)^2=4r^2,y=\sqrt{4r^2-4x^2}\]
\[s=surface~ area=2\pi x y+2\pi x^2=2\pi x \sqrt{4r^2-4x^2}+2\pi x^2\]
\[\frac{ ds }{ dx }=2\pi \left( x \frac{ -8x }{2\sqrt{4r^2-4x^2} }+1*\sqrt{4r^2-4x^2} \right)+4\pi x\]
\[=4\pi \left( \frac{ -x^2 }{\sqrt{r^2-x^2} }+\sqrt{r^2-x^2} \right)+4\pi x\]
\[=4\pi \left( \frac{ -x^2+r^2-x^2 }{\sqrt{r^2-x^2} } \right)+4\pi x\]