Let a cylinder
with radius OE = x inscribed in the given cone (Fig 6 18) The height QE of the cylinder
is given by
Rationalised 2023-24
APPLICATION OF DERIVATIVES
171
QE
OA  = EC
OC    (since ∆QEC ~ ∆AOC)
or
QE
h  = r
x
r
−
or
QE =
(
)
h r
x
r
−
Let S be the curved surface area of the given
cylinder Then
S  ≡ S(x) = 2
(
)
xh r
x
r
π
−
 = 
2
2
(
)
h rx
x
πr
−
or
2
S ( )
(
2 )
4
S ( )
h
x
r
x
r
h
x
r
π
 ′
=
−

− π
 ′′
=

Now S′(x) = 0 gives 
x =r2