We have to find the maximum volume of the cylinder.

Now let the height of the required cylinder by equal to h and let its radius be equal to r.

Now the radius of the base of the cone is 4 cm. And the altitude is 12 cm. So the angle made by the line placed along the side is arc tan ( 12 / 4)

For a cylinder to fit into the cone, we require that

arc tan( h/(4-r)= arc tan (12 / 4)

=> h / (4- r) = 12/4 = 3

=> h = 3(4-r)

=> h = 12 - 3r

Now the volume of the cylinder is pi*r^2*h

=> pi*r^2*(12 - 3r)

=> 12*pi*r^2 - 3*pi*r^3

To maximize 12*pi*r^2 - 3*pi*r^3, lets find the derivative

V = 12*pi*r^2 - 3*pi*r^3

V' = 24*pi*r - 9*pi*r^2

Equate V' to 0.

=> 24*pi*r - 9*pi*r^2 = 0

=> 24 - 9*r = 0

=> r = 24/9

=> r = 8/3

Therefore the radius of the cylinder should be 8/3.

The volume then is pi*r^2*h

=> pi*(8/3)^2*4

=> 89.36 cm^3

The volume of the required cylinder is 89.36 cm^3.