Find the volume of the larges cylinder that can be inscribed in a sphere of radius `r`

Explanation

In `triangle OAB`
`r^2=R^2+x^2.....(i)`
Volume of cylinder `=pi R^2h`
`=>V=piR^2 xx 2x`
`=>V=2piR^2x`
`=>V=2pi(r^2-x^2)x`
`=>V=2pi(r^2x-x^3)...(ii)`
`(dV)/(dx)=2pi(r^2-3x^2)`
and `(d^2V)/(dx^2)=2pi(-6x)`
for critical points,
`(dV)/(dr)=0`
`=>2pi(r^2-3x^2)=0`
`=>r^2=3x^2`
`=>x=r/sqrt3`
now,
`((d^2V)/(dx^2))_(x=r/sqrt3)=2pi(-6 xx r/sqrt3) lt 0`
so, volume of cylinder is maximum when
`x=r/sqrt3`
Therefore maximum volume of cylinder
`=2pi(r^2x-x^3)`
`=2pi(r^2 xx r/sqrt3-r^3/3sqrt3)`
`=2pi((3r^3-r^3)/(3sqrt3))`
`=(4pir^3)/(3sqrt3)`