A circular cylinder is to fit inside a sphere of radius 10 cm. Calculate the maximum possible volume of the cylinder. Use differentiation (dy/dx)It...

Let r a be the radius of the cylinder and 2h be the height
of the cylinder. and R be the radius of the sphere.

Then
the  consder  the points , O the centre of the sphere, the centre  P of the top cercular
face and a point A on the top cirular circumference .

OPA
is right angle OP= h, OA = R and PA = r.

Then R^2 =
h^2+r^2.........(1).

Volume of the cylinder  V = pir^2*2h =
pi(R^2-h^2)2h = 2pi (R^2*h-h^3)

For maximum volume, dV/dh =
0  for som h = h1 and d^2V/dh^2  = should be negative at h =
h1.

dV/dh= V'  = 0 gives: 2pi (2R^2*h-h ^3)' =
0

V' = 2pi (R^2-3h^2) = , (R^2-3h^2) = 0 , R^2 = 3h^2 , h =
(R^2/3)^(1/2) = R/sqrt3

d2V/dh^2 = 2pi(R^2-3h^2)' = - 12pih
is negative as h >0. So for h = R/sqrt3, V = 2pi(R^2-h^2)h attains the
maximum.

Therefore maximum volume V = 2pi (R^2-R^2/3)
(R/sqrt3)

= 2pi (2/3)R^2
(R/sqrt3)

=  (4/9)(sqrt3) Pi*
R^3

= 2418.4 cm^3.