If the curved surface area of a sphere is equal to the curved surface area of a hemisphere, find the radius of the hemisphere.

Question: If the curved surface area of a sphere is equal to the curved surface area of a hemisphere, find the radius of the hemisphere.

Solution:
Let R be the radius of the sphere and r be the radius of the hemisphere.

Curved surface area of sphere = Curved surface area of hemisphere
4πR² = 2πr²
⇒ 2R² = r²
⇒ r = √2 × R

∴ The radius of the hemisphere = √2 times the radius of the sphere.