A hemisphere and a right circular cone have the same volume and radius. Find the ratio of their curved surface areas (Hemisphere : Cone).
Show answer & explanation
Correct answer: A
Explanation
Let the radius of both be r.
Volume of the hemisphere = (2/3)πr³
Volume of the cone = (1/3)πr²h
The two volumes are equal:
(2/3)πr³ = (1/3)πr²h
2r = h
Height of the cone = 2r
Slant height of the cone:
l = √(r² + h²)
= √(r² + (2r)²)
= √5r
Curved surface area of the hemisphere = 2πr²
Curved surface area of the cone = πrl
= πr(√5r)
= πr²√5
Ratio = 2πr² : πr²√5 = 2 : √5