Right circular cone cut parallel to base — cone vs frustum volume ratio: A right circular cone of total height h is cut by a plane parallel to the base at a distance h/3 from the base. Find the ratio of the volumes of the smaller cone (toward the vertex) and the frustum.

Introduction / Context:A plane parallel to the base of a cone slices off a top cone similar to the original. Similarity gives linear scale; volumes then scale by the cube of the linear factor. The remainder is a frustum.

Given Data / Assumptions:

• Total cone height = h.
• Slice is at distance h/3 from the base, so the top small cone has height h − h/3 = 2h/3.
• Similarity ratio (small to original) = (2h/3)/h = 2/3.

Concept / Approach:For similar solids, volume ratio = (linear ratio)^3. Hence V_small / V_full = (2/3)^3 = 8/27. The frustum volume is V_full − V_small = (27/27 − 8/27)V_full = 19/27 V_full. Therefore, V_small : V_frustum = 8 : 19.

Step-by-Step Solution:Linear ratio k = 2/3Volume ratio small:full = k^3 = 8/27Frustum = 1 − 8/27 = 19/27Therefore, small:frustum = (8/27):(19/27) = 8:19

Verification / Alternative check:Pick any convenient dimensions satisfying similarity; the ratio remains 8:19.

Why Other Options Are Wrong:4/7, 6/19, 18/19 mis-handle cubic scaling or subtract from 1 incorrectly.

Common Pitfalls:Using area (square) scaling instead of volume (cube) scaling; misreading where h/3 is measured from.

Final Answer:8/19