Cylinder height from sphere volume equality: A circular cylinder has the same radius as a sphere and their volumes are equal. The height of the cylinder is equal to what multiple of its radius?
Aptitude
Volume and Surface Area
Difficulty: Easy
Choose an option
-
A4/3 times its radius
-
B2/3 times its radius
-
CEqual to the radius
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DEquals to its diameter
Answer
Correct Answer: 4/3 times its radius
Explanation
Introduction / Context:Equating cylinder and sphere volumes with common radius yields a direct relationship between cylinder height and radius. This is a standard mensuration identity.
Given Data / Assumptions:
- Common radius r.
- V_cylinder = π r^2 h; V_sphere = (4/3) π r^3; volumes equal.
Concept / Approach:
- Set π r^2 h = (4/3) π r^3 and solve for h in terms of r.
Step-by-Step Solution:
π r^2 h = (4/3) π r^3 ⇒ cancel π r^2 (r>0) to get h = (4/3) r.Verification / Alternative check:
Pick r=3 ⇒ sphere V = (4/3)π*27 = 36π; cylinder with h=4 gives V=π*9*4=36π, equal.Why Other Options Are Wrong:
- 2/3 r or r: Too short to produce equal volume.
- Diameter (2r): Too tall; volume would exceed the sphere volume.
Common Pitfalls:
- Dropping the 1/3 in sphere’s volume.
- Confusing diameter and radius when interpreting options.
Final Answer:
4/3 times its radius