A cone and a hemisphere have equal curved surface areas and equal radii. Find the ratio of their volumes.
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Correct answer: A
Explanation
Curved surface area of cone = πrℓ
Curved surface area of hemisphere = 2πr²
πrℓ = 2πr²
ℓ = 2r
for cone
ℓ² = r² + h²
(2r)² = r² + h²
4r² = r² + h²
h² = 3r²
h = √3r
Volume of the cone = (1/3)πr²h = (1/3)πr²(√3 r) = (√3/3)πr³
Volume of hemisphere = (2/3)πr³
Cone : Hemisphere = (√3/3)πr³ : (2/3)πr³= √3 : 2