If the height , curved surface area and volume of a right circular cone be h, c and v respectively , then that `3pi vh^3-c^2h^2+9v^2=0`.

Explanation

Let the radius of base and slant height of the cone be r unit and l unit respecitvely.
`therefore l^2=h^2+r^2`....................(1)
Again, `v=(1)/(3)pi r^2h` .................(2) and `c=pi r l`................(3)
`therefore 3 pi vh^3-c^2h^2+9v^2`
`=3pi xx(1)/(3)pi r^2h^3-(pi r l)^2xxh^2+9xx((1)/(3)pi r^2h)^2`[by (2) and (3)].
`=pi^2r^2h^4-pi^2r^2l^2h^2+pi^2r^4h^2`
`=pi^2r^2h^4-pi^2r^2(r^2+h^2).h^2+pi^2r^4h^2`[by (1)].
`= pi ^2r^2h^4-pi^2r^4h^2-pi^2r^2h^4+pi^2r^4h^2=0`
Hence `3pi vh^3-c^2h^2+9v^2=0` .