A sphere and a right circular cylinder have the same radius and equal curved surface areas. Find the ratio of their volumes (sphere : cylinder).
Aptitude
Volume and Surface Area
Difficulty: Medium
Choose an option
-
A3 : 4
-
B2 : 3
-
C3 : 2
-
D4 : 3
Answer
Correct Answer: 2 : 3
Explanation
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:
h = 2r from CSA equalityV_sphere = (4/3)πr^3; V_cyl = 2πr^3Volume ratio = (4/3)πr^3 : 2πr^3 = 4/3 : 2 = 2 : 3Verification / 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