Question

Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.

Answer 1

Let, r be the radius and h be the height of the cylinder.

Let S be the surface area of the cylinder,

S=2π r 2 +2πrh S−2π r 2 =2πrh h= S−2π r 2 2πr

Let, V be the volume of cylinder,

V=π r 2 h =π r 2 ( S−2π r 2 2πr ) =r( S−2π r 2 2 ) = Sr 2 −π r 3

Differentiate volume with respect to r,

V ′ = S 2 −3π r 2 (1)

Put V ′ =0,

S 2 −3π r 2 =0 S 2 =3π r 2 r 2 = S 6π r= S 6π

Differentiate equation (1) with respect to r,

V ″ ( r )=−6πr V ″ ( S 6π )=−6π( S 6π ) <0

This shows that V ″ is negative, so, the volume is maximum when r= S 6π .

At r 2 = S 6π , the height of the cylinder becomes,

h= 6π r 2 −2π r 2 2πr = 4π r 2 2πr =2r

Therefore, it is proved that the volume of cylinder is maximum when h=2r.