Two cones — volumes 1 : 3 and diameters 3 : 5 — find heights ratio: The volumes of two right circular cones are in the ratio 1 : 3 and their diameters are in the ratio 3 : 5. Find the ratio of their heights.

Introduction / Context:
Using proportionality V ∝ r^2 h, a known ratio of volumes and radii (or diameters) lets us solve for the ratio of heights. Since diameter and radius scale the same, we can use the diameter ratio directly to infer the radius ratio.

Given Data / Assumptions:

• V1 : V2 = 1 : 3.
• D1 : D2 = 3 : 5 ⇒ r1 : r2 = 3 : 5.

Concept / Approach:
Set (r1^2 h1) : (r2^2 h2) = 1 : 3. Substitute r1^2 : r2^2 = 9 : 25, then solve for h1 : h2.

Step-by-Step Solution:
(9 h1) : (25 h2) = 1 : 3Cross-multiply: 3 * 9 h1 = 25 h2 ⇒ h1 : h2 = 25 : 27

Verification / Alternative check:
Choose r1 = 3, r2 = 5 and heights in 25 : 27; then V1 : V2 ∝ (9*25) : (25*27) = 225 : 675 = 1 : 3.

Why Other Options Are Wrong:
They ignore the square on r or put the ratios in the wrong order.

Common Pitfalls:
Using r instead of r^2 in the proportional relation to volume.

Final Answer:
25 : 27