The densities of two solid spheres  A and B  of the same radii  R vary with radial distance  r as  ρAr=krR and  ρBr=krR5, respectively, where k  is a constant.  The moments of inertia of the individual spheres about axes passing through their centers are  IA and  IB, respectively.  If  IBIA=n10, the value of  n is

At a distance of r, select a hollow sphere of thickness dr and mass dm. Moment of Inertia of this elemental Hollow sphere is given by, dl=23(dm)r2  here dm=ρdV=ρ4πr2dr⇒dl=23ρ4πr2drr2 For  Sphere A:ρ=krRdl=23krR4πr2drr2⇒IA=8kπ3R∫0R r5dr⇒IA=4kπR59 For Sphere B:ρ=krR5dI=23kr5R54πr2drr2⇒IB=8kπ3R5∫0R r9dr⇒IB=4kπR515 Ratio =IBIA=4kπR5154kπR59=35=610 So, n=6