Shortcut Trick Equate total volumes: 24 × (2/3) × r^(3) = R^(2) × H (π cancels on both sides). 16 × r^(3) = 12 × 12 × 24 r^(3) = 144 × (24 ÷ 16) = 144 × 1.5 r^(3) = 216 ⇒ r = 6 cm. ∴ The correct answer is 6 cm. Alternate Method Given: Number of solid hemispheres (n) = 24 Radius of cylinder (R) = 12 cm Height of cylinder (H) = 24 cm Formula Used: Volume of Hemisphere = (2/3) × π × r^(3) Volume of Cylinder = π × R^(2) × H Calculations: ⇒ Total volume of hemispheres = Volume of cylinder ⇒ 24 × (2/3) × π × r^(3) = π × R^(2) × H ⇒ 16 × π × r^(3) = π × 12^(2) × 24 ⇒ 16 × r^(3) = 144 × 24 ⇒ r^(3) = (144 × 24) ÷ 16 ⇒ r^(3) = 144 × 1.5 ⇒ r^(3) = 216 ⇒ r = ^(3)√216 = 6 cm ∴ The correct answer is 6 cm. Related Concepts: Volume of Sphere The volume of a complete solid sphere is given by V = (4/3) × π × r^(3). Surface Area of Hemisphere The Curved Surface Area (CSA) is 2πr^(2) and the Total Surface Area (TSA) is 3πr^(2). Volume Conservation Rule When one solid is melted and recast into another shape, the total volume remains constant, assuming no material loss. Cylinder Surface Area The CSA of a cylinder is 2πRH and its TSA is 2πR(R + H).