A sphere and a right circular cylinder have the same radius and equal curved surface areas. Find the ratio of their volumes (sphere : cylinder).

Introduction / Context:
Relate the curved surface areas (CSA) to tie the cylinder’s height to the common radius, then compute volumes and form the ratio sphere:cylinder.

Given Data / Assumptions:

• CSA_sphere = 4πr^2.
• CSA_cylinder = 2πrh (lateral area).
• Equal CSAs and same radius r.

Concept / Approach:
Set 4πr^2 = 2πrh ⇒ h = 2r. Then V_sphere = (4/3)πr^3 and V_cyl = πr^2h = 2πr^3. Take the ratio.

Step-by-Step Solution:

Verification / Alternative check:
π and r^3 cancel; ratio simplifies exactly to 2:3.

Why Other Options Are Wrong:
They mismatch the derived height h = 2r and corresponding cylinder volume.

Common Pitfalls:
Using total surface area instead of curved area; confusing diameter with radius in the height relation.

Final Answer:
2 : 3