Q4 of 56 Page 13

A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. The ratio of the volume of the smaller cone to the whole cone is (CBSE 2012)


Given: A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base


Let ‘H’ be the height of the cone.


Let ‘R’ be the Radius of the complete cone.


Volume of a cone is given by: πr2h


Here,


AB = BD =


Let r be the radius of the smaller cone.


∴ In ΔABC and ΔADE


∠ABC = ADE (90°)


∠CAB = ∠EAB (common)


∴ ΔABC ΔADE (AA similarity criterion)


⇒ = (Corresponding sides are proportional)


⇒ =


⇒ R = 2r


Volume of smaller cone = π(r)2 × h = π(BC)2 × AB = π(r)2 × = cm3


Volume of whole cone = π(r)2 × h = π(DE)2 × AD = π(2r)2 × H = πr2H cm3


∴ = =


∴ The ratio of the volume of the smaller cone to the whole cone is 1:8

More from this chapter

All 56 →
2

A sphere of diameter 18 cm is dropped into a cylinder vessel of diameter 36 cm, partly filled with water. If the sphere is completely submerged then the water level rises by (CBSE 2011)

 3

A solid is hemispherical at the bottom and conical (of same radius) above it. If the surface areas of the two parts are equal then the ratio of its radius and the slant the height of the conical part is (CBSE 2011)

 5

If the radius of the base of a right circular cylinder is halved, keeping the height the same. Then the ratio of the volume of the cylinder thus obtained to the volume of original is (CBSE 2012)

 6

The radii of the circular ends of a bucket of height 40 cm are 24 cm and 15 cm. The slant height (in cm) of the bucket is (CBSE 2012)