Two solid hemispheres of radii R and 2R are melted to form a sphere and a cone. The cone has the same radius as the new sphere, and its height is twice the sphere's radius. Find the radius of the newly formed sphere.
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Correct answer: A
Explanation
Let the radius of the new sphere be r.
Radius of the cone = r and height = 2r.
Total volume of both hemispheres = (2/3)πR³ + (2/3)π(2R)³
= (2/3)πR³ + (16/3)πR³
= 6πR³
Volume of the new sphere = (4/3)πr³
Volume of the cone = (1/3)πr²(2r)
= (2/3)πr³
Therefore, total new volume = (4/3 + 2/3)πr³
= 2πr³
By the conservation of volume,
6πR³ = 2πr³
R³ = 3r³
R = ∛3 × r