Cylinders (equal volumes) — Two cylinders have equal volume and heights in ratio 1:2. What is the ratio of their radii?

Introduction / Context:
Equal volumes with different heights imply different radii. Since V = π r^2 h, the radius must compensate inversely with the square root.

Given Data / Assumptions:

• V1 = V2
• h1:h2 = 1:2

Concept / Approach:
From π r1^2 h1 = π r2^2 h2 ⇒ r1^2 / r2^2 = h2 / h1 ⇒ r1 / r2 = √(h2/h1) = √2.

Step-by-Step Solution:
r1:r2 = √2:1

Verification / Alternative check:
Test with h1=1, h2=2, r1=√2, r2=1 gives equal volumes.

Why Other Options Are Wrong:
1:√2 flips the correct ratio; 1:2 or 1:4 treat radius linear with height, not square root.

Common Pitfalls:
Missing the square on r in V = πr^2h.

Final Answer:
√2 : 1