Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius `R` is `(2R)/(sqrt(3))` .

Explanation

`h^2=R^2-r^2`
Volume of cylinder
`r^2=R^2-h^2`
`pir^2(2h)=2pir^2h`
`2pi(R^2-h^2)h`
`(dv)/(dh)=0`
`d/dx[2pi(R^2-h^2)h]=0`
`(R^2-h^2)+h*(-2h)=0`
`R^2-3h^2=0`
`h=R/sqrt3`
Height of the cylinder at maximum
`H=2h=(2R)/sqrt3`
`V_(max)=2piR^2h=2pi(R^2-h^2)h`
`=2pi(R^2-R^2/3)R/sqrt3`
`=2pi(2R)^2/3*R/sqrt3`
`(4piR^3)/(3sqrt3)`
hence, the height of
the cylinder of maximum volume that can be inscribed in a sphere of radius `R`
is `(2R)/(sqrt(3))`