The height of a right circular cylinder of maxium volume inscirbed in a sphere of radius 3 is

Options

A. `sqrt(6)`
B. `2sqrt(3)`
C. `sqrt(3)`
D. `(2)/(3)sqrt(3)`

Answer

B

Explanation

Let a sphere of radius, 3, which inscribed a right circular cylinder having radius r and height is h, so
From the figure, `(h)/(2)= 3 cos theta`
`rArr h=6 cos theta`
and `r=3sin theta " "...(i)`

For maxima or minima, `(dV)/(dtheta)=0`
`rArr 54 pi[ 2 sin theta cos^(2) theta- sin ^(3) theta]=0`
`rArr sin theta[2 cos^(2) theta-sin^(2) theta]=0`
`rArr tan^(2) theta=2 " [ :. theta in (0,(pi)/(2))]`
`rArr tan theta=sqrt(2)`
` rArr sin theta=sqrt((2)/(3)) and cos theta=(1)/(sqrt(3))" "...(ii)`
From Eqs.(i) and (ii), we get
`h=6(1)/(sqrt(3))=2 sqrt(3)`