Two cones — deduce height ratio from volume and radius ratios: The volumes of two cones are in the ratio 2 : 3 and the radii of their bases are in the ratio 1 : 2. Find the ratio of their heights.

Introduction / Context:
For cones, volume is proportional to r^2 h. Given the ratio of volumes and radii, one can solve for the ratio of heights using proportional reasoning, without computing any absolute values.

Given Data / Assumptions:

• V1 : V2 = 2 : 3.
• r1 : r2 = 1 : 2.
• V ∝ r^2 h for fixed proportionality constant (1/3)π.

Concept / Approach:
Set (r1^2 h1) : (r2^2 h2) = 2 : 3. Substitute r1^2 : r2^2 = 1^2 : 2^2 = 1 : 4 and solve for h1 : h2.

Step-by-Step Solution:
(1 * h1) : (4 * h2) = 2 : 3Cross-multiply: 3h1 = 8h2 ⇒ h1 : h2 = 8 : 3

Verification / Alternative check:
Try concrete numbers, e.g., r1 = 1, r2 = 2; pick h1 = 8, h2 = 3. Then V1 ∝ 1*8=8 and V2 ∝ 4*3=12 → 8:12 = 2:3 as required.

Why Other Options Are Wrong:
They either invert the ratio or ignore the square on radius in the volume expression.

Common Pitfalls:
Using r instead of r^2 when connecting volume ratio to height ratio.

Final Answer:
8 : 3