Cylinder melted into a cone (same radius)\nA cylinder of radius 4 cm and height 8 cm is melted and recast into a cone of the same radius. Find the height of the cone (in cm).

Introduction / Context:
Melting and recasting conserves volume. With equal radii, we equate the cylinder’s volume to the cone’s volume and solve for the cone’s height directly.

Given Data / Assumptions:

• Cylinder radius r = 4 cm, height H_cyl = 8 cm
• Cone radius r = 4 cm, height H_cone = ?
• Volumes: V_cyl = π r^2 H_cyl; V_cone = (1/3) π r^2 H_cone

Concept / Approach:
Set π r^2 H_cyl = (1/3) π r^2 H_cone. Since r is the same, π r^2 cancels, yielding a simple proportionality between heights.

Step-by-Step Solution:

Verification / Alternative check:
Compute volumes numerically: cylinder = 128π; cone with H = 24 gives (1/3)*16π*24 = 128π—exactly equal.

Why Other Options Are Wrong:
12, 36, and 48 correspond to one-half, 1.5x, or 2x the correct value; only 24 preserves equal volume with same radius.

Common Pitfalls:
Forgetting the 1/3 factor in cone volume or not cancelling common factors leads to algebra mistakes.

Final Answer:
24 cm