a sphere is melted and recast into a hemisphere show that the ratio of surface area of square to that of hemisphere is 4:3³√4
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Answers
Given : A sphere is melted and recast into a hemisphere.
To Find : show that ratio of surface area of sphere to that of hemisphere is 4:3³√4
Solution:
Let say Radius of Sphere = R
Then Volume = (4/3)πR³
Surface area of sphere = 4πR²
recast into a hemisphere radius = r
=> Volume = (2/3)πr³
Surface area of hemisphere = 3πr²
Volume of both would be same
(2/3)πr³ = (4/3)πR³
=>r³ = 2R³
=> r = ∛2 R
Surface area of hemisphere = 3πr² = 3π ( ∛2 R)²
= 3π∛4 R²
Surface area of sphere / surface area of hemisphere = 4πR² / 3π∛4 R²
= 4 : 3 ∛4
QED
Hence proved
ratio of surface area of sphere to that of hemisphere = 4 : 3 ∛4
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