Cones — Volumes in ratio 1:4; diameters in ratio 4:5. Find the ratio of their heights.

Introduction / Context:
For cones, V ∝ r^2 h. With diameter ratio 4:5, radius ratio is 2:2.5 = 4:5. Use the volume ratio to solve for height ratio.

Given Data / Assumptions:

• V1:V2 = 1:4
• r1:r2 = 4:5 ⇒ r1^2:r2^2 = 16:25

Concept / Approach:
(r1^2 h1):(r2^2 h2) = 1:4 ⇒ (16 h1):(25 h2) = 1:4 ⇒ h1/h2 = (1/4) * (25/16) = 25/64.

Step-by-Step Solution:
h1:h2 = 25:64

Verification / Alternative check:
Check by plugging into V ∝ r^2h.

Why Other Options Are Wrong:
They ignore r^2 or invert ratios.

Common Pitfalls:
Using diameter directly without squaring radius in volume.

Final Answer:
25 : 64