The height of right circular cylinder of maximum volume in a sphere of diameter `2a` is

Options

A. ` 2sqrt3 a `
B. ` sqrt 3 a `
C. ` ( 2a ) /( sqrt 3 ) `
D. ` ( a ) /( sqrt 3 ) `

Answer

C

Explanation

Let the radius and height of the cylinder are r and h, respectively.

In ` Delta AOM`,
` r^ 2 + (( h^ 2 )/( 4 ) ) = a^ 2 `
` therefore h ^ 2 = 4 ( a ^ 2 - r ^ 2 ) `
Now, ` V = pi r ^ 2 h = pi ( a ^ 2 h - ( 1 )/( 4 ) h^ 3 ) `
For max or min,
` (dV) / ( dh ) = pi ( a ^ 2 - ( 3)/( 4 ) h ^ 2 ) = 0 `
` rArr h = ( ( 2 ) /( sqrt 3 )) a `
Now, ` ( d ^ 2 V ) /( dh^ 2 ) = - ( 6h ) /( 4 ) lt 0 `
So, V is maximum at ` h = ( 2a )/( sqrt3 ) `