Melted hemisphere forms a cylinder – Given CSA:TSA = 2:3, hemisphere radius 14 cm: A solid hemisphere of radius 14 cm is melted and cast into a cylinder. For the cylinder, curved surface area : total surface area = 2 : 3. Find the cylinder’s base radius (in cm).

Introduction / Context:Conservation of volume applies when reshaping solids. For a cylinder with CSA:TSA = 2:3, a fixed relation between height and radius follows. Combine with volume equality from the original hemisphere.

Given Data / Assumptions:

• Hemisphere radius R = 14 cm.
• For cylinder: CSA = 2πrh; TSA = 2πrh + 2πr^2; ratio 2:3.

Concept / Approach:From CSA:TSA = 2:3 ⇒ h : (h + r) = 2 : 3 ⇒ h = 2r. Volume equality: hemisphere volume (2/3)πR^3 = cylinder volume πr^2h = πr^2(2r) = 2πr^3.

Step-by-Step Solution:

Verification / Alternative check:Then h = 2r ≈ 19.42 cm; check CSA:TSA = 2:3 by substitution holds algebraically.

Why Other Options Are Wrong:10.33, 7.33, 12.33 cm deviate from the exact cube-root result. The originally printed choices were malformed; the repaired set includes the accurate value.

Final Answer:9.71 cm