The volume of a right circular cone is 9856 cm 3 and the area of its base is 616 cm 2 . Find the slant height of the cone.

Use the given values to find the radius ' r ' and slant height ' l ', Given: Area of the base = 616 cm 2 π r 2 = 616 22 7 × r 2 = 616 r 2 = 616 × 7 22 r 2 = 196 ⇒ r = 196 = 14 Substitute the value of r in the formula used to find out the volume of the cone, Volume of the cone = 1 3 π r 2 h 9856 = 1 3 π r 2 h 9856 = 1 3 × a r e a × h h = 9856 × 3 616 h = 48 To find the slant height of the cone by substituting the values of ' r ' and ' h ', Slant height, l = h 2 + r 2 l = 48 2 + 14 2 l = ( 2304 + 196 ) l = ( 2500 ) = 50 Hence, the slant height of the cone is 50 cm .

The volume of a right circular cone is 9856 cm 3 and the area of its base is 616 cm 2 . Find total surface area of the cone.

Given : For circular cone : Volume of cone = 9856 cm 3 Area of the base = 616 cm 2 Radius r = Area of base π = 616 × 7 22 = 28 × 7 = 196 = 14 cm Volume = 1 3 π r 2 h ⇒ 1 3 × 22 7 × 14 × 14 × h = 9856 cm 3 ⇒ h = 9856 × 3 × 7 22 × 14 × 14 = 48 cm Slant height l = r 2 + h 2 = 14 2 + 48 2 = 196 + 2304 = 2500 = 50 cm Total surface area = π r l + r = 22 7 × 14 50 + 14 = 44 64 = 2816 cm 2 Hence, the total surface area is 2816 cm 2 .

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