Question

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is . Also find the maximum volume.

Answer 1

Let, R be the radius of the sphere, h be the height and x be the diameter of the cylinder.

Use the Pythagoras theorem in ΔABC,

( CB ) 2 + ( AB ) 2 = ( AC ) 2 h 2 + x 2 = ( R+R ) 2 x 2 =4 R 2 − h 2 (1)

The volume of cylinder is,

V=π ( x 2 ) 2 ×h V=π( 4 R 2 − h 2 4 )×h V=4πh R 2 − π h 3 4

Differentiate above equation with respect to h,

dV dh =4π R 2 d( h ) dh − π 4 d( h 3 ) dh =π R 2 − 3π 4 h 2

Equate derivative of the volume is zero.

dV dh =0 π R 2 − 3π 4 h 2 =0 h= 2R 3

The value of d 2 V d h 2 ,

dV dh =π R 2 − 3π 4 h 2 d 2 V d h 2 = −3πh 4

Substitute the value of h= 2R 3 ,

d 2 V d h 2 = −3πh 4 d 2 V d h 2 = −3π( 2R 3 ) 4 <0

So, volume is maximum when h= 2R 3 .

From equation (1),

x 2 =4 R 2 − h 2 x 2 =4 R 2 − ( 2R 3 ) 2 x 2 = 8 3 R 2

The maximum value of the volume is,

V=π ( x 2 ) 2 h

Substitute the value of x 2 and h.

V=π ( x 2 ) 2 h = 16π R 3 12 3 = 4π R 3 3 3  cubic unit

Thus, the maximum volume is 4π R 3 3 3  cubic unit.