The height of a cone is 30 c m . A small cone is cut off at the top by a plane parallel to its base. If its volume be 1 27 of the volume of the given cone, at what height above the base is the section cut?

Height of a cone = 30 c m A small cone is cut off from the top of the cone Therefore, Volume of small cone = 1 27 of the volume of the given cone. Volume of given cone = 1 3 π R 2 H and volume of small cone = 1 3 π r 2 h So, 1 3 π r 2 h = 1 27 × 1 3 π R 2 H r 2 R 2 h = 1 27 × 30 r 2 R 2 h = 10 9 . . . . . . ( i ) But r R = h H = h 30 ⇒ r R = h 30 . . . . . ( i i ) From i and i i h 30 2 h = 10 9 ⇒ h 2 × h 900 = 10 9 h 3 = 10 × 900 9 = 1000 = 10 c m A line parallel to base at a distance of 30 - 10 = 20 c m

A solid metallic sphere of radius 6 c m is melted and made into a solid cylinder of height 32 c m . Find the curved surface area of the cylinder. (Take π = 3 . 1 )

Radius of metallic sphere R = 6 c m Height of cylinder h = 32 c m Volume of cylinder = Volume of metallic sphere π r 2 h = 4 3 π R 3 r 2 × 32 = 4 3 × 6 3 r 2 = 4 × 6 × 6 × 6 3 × 32 r 2 = 9 r = 3 c m Then, curved surface area of the cylinder = 2 π r h = 2 × 3 . 1 × 3 × 32 = 595 . 2 c m 2