The volumes of a sphere and a right circular cylinder (with the same radius) are equal. Find the ratio of the diameter of the sphere to the height of the cylinder.
Aptitude
Volume and Surface Area
Difficulty: Easy
Choose an option
-
A1 : 2
-
B2 : 1
-
C2 : 3
-
D3 : 2
Answer
Correct Answer: 3 : 2
Explanation
Introduction / Context:Equate volumes of sphere and cylinder with common radius r. Express the cylinder height in terms of r, then form the ratio of sphere diameter to that height.
Given Data / Assumptions:
- V_sphere = (4/3)πr^3.
- V_cyl = πr^2h.
- Equal volumes: V_sphere = V_cyl.
Concept / Approach:Solve (4/3)πr^3 = πr^2h for h, then compute (diameter):(height).
Step-by-Step Solution:
(4/3)πr^3 = πr^2h ⇒ h = (4/3)rDiameter of sphere = 2rRatio = 2r : (4/3)r = 2 : 4/3 = 3 : 2Verification / Alternative check:Cancel π and r^2 safely since r > 0; ratio reduces cleanly.
Why Other Options Are Wrong:They do not reflect h = 4r/3 and D = 2r.
Common Pitfalls:Using circumference or surface areas instead of volumes; forgetting diameter is 2r.
Final Answer:3 : 2