Creating a frustum, a right circular cone is sliced parallel to its base,. The height of the original cone is 30 cm, and the cut is made 10 cm from the vertex. What is the ratio of the volumes of the frustum to that of the entire cone?

Correct — A. The cut is parallel to the base, so the piece above it is a small cone similar to the original.

Height ratio, small : whole = 10 : 30 = 1 : 3
Volume ratio = cube of the height ratio = 1³ : 3³ = 1 : 27

Frustum = whole − small cone = 27 − 1 = 26 parts
Frustum : whole cone = 26 : 27 → option (a)

Why the others are wrong

• (b)
  11:18 — 11 : 18 leaves the top cone 7⁄18 of the volume. That needs a height ratio of ∛(7⁄18) ≈ 0.73, a cut about 22 cm from the vertex, not 10 cm.
• (c)
  16:19 — 16 : 19 leaves the top cone 3⁄19 of the volume. That needs a height ratio of ∛(3⁄19) ≈ 0.54, a cut about 16 cm from the vertex, not 10 cm.
• (d)
  14:15 — 14 : 15 leaves the top cone 1⁄15 of the volume. That needs a height ratio of ∛(1⁄15) ≈ 0.41, a cut about 12 cm from the vertex, not 10 cm.

A plane parallel to the base cuts off a smaller cone similar to the whole: its radius and height shrink by the same factor k.

Volume = (1⁄3)πr²h multiplies three lengths, so the small cone's volume is k³ times the whole. Here k = 1⁄3 and k³ = 1⁄27.

The frustum is what remains: 1 − k³ of the cone, here 26⁄27.

Key facts

• Volume of a cone = (1⁄3)πr²h.
• Similar solids with lengths in ratio k have volumes in ratio k³.
• A cone cut one third of the way down from its vertex leaves a frustum of 26⁄27 of its volume.

Common traps

• Using the height ratio 1 : 3 directly, which gives a frustum of 2⁄3 instead of 26⁄27.
• Measuring the cut from the base instead of the vertex, which changes k from 1⁄3 to 2⁄3.
• Answering small cone : frustum (1 : 26) when the question asks frustum : whole cone.

No directly related past PYQ was found.