Question

A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. What is the ratio of the volume of the smaller cone to the whole cone?

(a) 1:4
(b) 1:2
(c) 1:3
(d) 1:8

Answer 1

The volume of the smaller cone that results from cutting a right circular cone at the midpoint of its height is 1/8 of the volume of the original cone. Hence, the ratio of the volume of the smaller cone to the whole cone is 1:8, making the correct answer (d) 1:8.

Step-by-step explanation:

The question pertains to finding the ratio of the volumes of two similar geometric shapes, in this case, the smaller cone that is formed when a right circular cone is cut parallel to its base at its midpoint, and the original cone itself. The volume of a cone is given by the formula V = (1/3)πr²h, where r is the radius and h is the height. Upon splitting the cone into two equal heights, we can observe that the radius of the smaller cone will be half of the original.

By substituting the new values into the volume formula, we find the volume of the smaller cone to be Vsmall = (1/3)π(r/2)²(h/2). Simplifying this yields Vsmall = (1/8)(1/3)πr²h, showing that the volume of the smaller cone is 1/8 of the original cone. Hence, the ratio of the volume of the smaller cone to the whole cone is 1:8.

In conclusion, when a solid right circular cone is cut into two equal parts by height, the volume of the smaller cone is an eighth of the whole cone. Therefore, the correct option answer is (d) 1:8.