Question

A right circular cone of height $h$ is cut by a plane parallel to the base and at a distance $h/3$ from the base. Then the volumes of the resulting cone and the frustum are in the ratio:

• A. $1 : 3$
• B. $8 : 19$
• C. $1 : 4$
• D. $1 : 7$

Hint:

For cones cut parallel to the base, use similar triangles to get height ratio, then cube it to get volume ratio before subtracting to find the frustum's volume.

Answer 1

The Correct Option is B

We have a cone of total height $h$. It is cut by a plane parallel to the base at a distance $h/3$ from the base.
That means the smaller cone on top has height = $h - h/3 = 2h/3$.
When two cones are similar, the ratio of their volumes is equal to the cube of the ratio of their heights (or radii).
Height ratio (smaller cone : original cone) = $(2h/3) : h = 2/3$.
Volume ratio (smaller cone : original cone) = $(2/3)^3 = 8/27$.
Therefore, volume of frustum = Volume of original cone $-$ Volume of smaller cone.
So frustum volume = $27 - 8 = 19$ (parts in ratio terms).
Thus, smaller cone : frustum = $8 : 19$.