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

Options

A. ` sqrt6 `
B. `sqrt 3 `
C. ` ( 2 ) / (3 ) sqrt 3 `
D. `2 sqrt 3 `

Explanation

` h = 2 (3 cos theta ) `
` r = 3 sin theta , v = pi r ^ 2 h , = pi 9 sin ^ 2 theta . 6 cos theta `
` V = 54 pi sin 2 theta cos theta , (dv ) / ( d theta ) = 0 `
` rArr 2 sin theta cos ^ 2 theta - sin ^ 3 theta = 0 rArr 2s ( 1 - s ^ 2 ) - s ^ 3 = 0 rArr 2 s - 2s ^ 3 - s ^ 3 = 0 `
` rArr 2 s - 3s ^ 3 = 0 rArr s = 0 or 2 - 3s ^ 2 = 0 `
` s = pm sqrt ((2 ) /(3)) therefore cos theta = sqrt( 1 - ( 2 ) /(3 ) ) = (1 ) /(sqrt 3 ) `
` h = 6 ((1 )/(sqrt3 )), h = 2 sqrt 3 `