A right circular cone having height of 30 cm is cut by two parallel planes at heights 10 cm and 20 cm from the base. What is the ratio of the volumes of the three parts (from top to bottom)?

Correct — D. The small cones share the apex, so measure every height from the top.

Cuts at 20 cm and 10 cm from the base sit 10 cm and 20 cm below the apex.
Cones above each level: heights 10 : 20 : 30 = 1 : 2 : 3

Cube the height ratio: volumes 1 : 8 : 27
Subtract to get the pieces: 1, 8 − 1 = 7, 27 − 8 = 19
Top : middle : bottom = 1 : 7 : 19 → option (d)

Why the others are wrong

• (a)
  8 : 19 : 27 — 8 : 19 : 27 mixes cones with slices: 8 is the 20 cm cone above the lower cut and 27 the whole cone. The three pieces must add to 27, and these add to 54.
• (b)
  1 : 6 : 20 — 1 : 6 : 20 does add to 27, but the split is wrong. The middle slice is 2³ − 1³ = 7 and the bottom slice 3³ − 2³ = 19.
• (c)
  8 : 19 : 64 — 8 : 19 : 64 gives the bottom piece 64 units, more than the whole cone. The cone is three 10 cm units tall, so its full volume is 27 units.

A plane parallel to the base cuts off a smaller cone similar to the original. Every length scales by the same factor k, so the volume scales by k³.

That is why heights are measured from the apex: every small cone shares it. The two lower pieces are frustums, and each one's volume is the difference of two cones.

Key facts

• Similar solids: volume ratio = (length ratio)³.
• Cutting a cone at equal height steps from the apex gives pieces 1 : 7 : 19 : 37, the differences of consecutive cubes.
• A cut at height h above the base of a cone of height H leaves a top cone of height H − h.

Common traps

• Stopping at 1 : 8 : 27, the cumulative cones, instead of subtracting to get the slices
• Measuring the small cones from the base, when they share the apex

No directly related past PYQ was found.