If the height of a cone of greatest volume that can be inscribed in a sphere of radius R is kR , then ratio of the volume of the cone to the volume of the sphere is

Step-by-step solution

aligned & In O D C , & r^2=R^2-(h-R)^2 & =2 h R-h^2 aligned Hence volume of cone A B C= 1 3 r^2 h V = 1 3 (2 h R-h^2 ) h= 1 3 (2 R h^2-h^3 ) Now for volume to be maximum, aligned & d V d h =0 & 1 3 [4 h R-3 h^2 ]=0 & h 0 Hence h= 4 R 3 & at h= 4 R 3 , d^2 V d h^2 < 0 aligned Hence at h= 4 R 3 , volume of cone is maximum. aligned & Volume of cone Volume of sphere = 1 3 r^2 h 4 3 R^3 = (2 h R-h^2 ) h 4 R^3 & = ( 8 R^2 3 - 16 R^2 9 ) 4 R 3 4 R^3 = 8 27 aligned