Equal radius, equal volume — cylinder vs cone: A right circular cylinder and a right circular cone have the same radius and the same volume. Find the ratio of the height of the cylinder to the height of the cone.
Correct Answer: 1 : 3
Introduction / Context: For solids with equal radius, comparing volumes quickly leads to a proportional relation between their heights. This question leverages the standard formulas for cylinder and cone volumes, focusing on their height relationship at equal volumes.
Given Data / Assumptions:
- Common radius r for both solids.
- Volumes are equal.
Concept / Approach: Volume of cylinder = π r^2 * h_cyl. Volume of cone = (1/3) * π r^2 * h_cone. Setting these equal and cancelling common terms yields a simple ratio between heights.
Step-by-Step Solution:
π r^2 * h_cyl = (1/3) * π r^2 * h_cone.Cancel π r^2 (nonzero) ⇒ h_cyl = (1/3) * h_cone.Therefore, h_cyl : h_cone = 1 : 3.Verification / Alternative check: If h_cone = 30, then h_cyl = 10. Substituting gives equal volumes, confirming the ratio.
Why Other Options Are Wrong: 3 : 1 inverts the relation; 3 : 5 and 2 : 5 suggest mismatched formulas; 1 : 2 does not satisfy the equality of volumes for equal radii.
Common Pitfalls: Forgetting the 1/3 factor in the cone’s volume; mixing radius equality with diameter or slant height.
Final Answer: 1 : 3