A circular cylinder is to fit inside a sphere of radius 10 c

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 is probably best to take as your independent variable the height, or half the height, of the cylinder)

Thanks in advance.

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.