The radius of the cone is 1.414(Sqrt of 2) times of height of the cone. a cube of max volume is cut from the same cone. find the ratio of volume of the cone to the volume of cube.

The volume of the cube will be maximum when the cube that is cut completely fits inside the cone. So, Consider the cube that completely fits the cone. Let the side of the cube be x. Since r/h=v2 when h=h-x r=v2(h-x) The diameter at this point should be equal to the height `x` So 2v2(h-x)=x we get x=2v2h/(1+2v2) thus ratio of volumes= pi(1+2v2)^3/24v2=5.19

If the cube has side 2x then the cross-section has height 2x and width 2xv2. If the height and base radius of the cone are h and hv2 then using similar triangles,
h / hv2 = 2x / (hv2 - xv2),
from which it follows that
(h - x) = 2x
h = 3x.
Check on height: we must have h>2x in this orientation, which is the case.
The volume of the cone is (1/3)pr^2h = (2/3)ph^3 = (2/3)p(3x)^3 and the volume of the cube is (2x)^3, so the ratio of volumes is (2/3)p(3/2)^3=p(3/2)^2=2.25p.