Height ratio for equal volumes (cylinder vs cone): A right 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.

Introduction / Context:
With the same radius, equal volumes force a relationship between heights using V_cyl = π r^2 h_c and V_cone = (1/3) π r^2 h_k. Equate and solve for h_c : h_k.

Given Data / Assumptions:

• Same radius r; equal volumes.
• V_cyl = π r^2 h_c, V_cone = (1/3) π r^2 h_k.

Concept / Approach:

• Set π r^2 h_c = (1/3) π r^2 h_k ⇒ h_c = h_k / 3.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 3:1 / 3:5 / 2:5: Inconsistent with the cone’s 1/3 volume factor.

Common Pitfalls:

• Inverting the ratio (writing 3:1 rather than 1:3).

Final Answer:

1 : 3