The radius of a solid sphere and a solid right circular cone are equal and the height of the cone is equal to the
diameter of its base. They are melted and the combined matter is recast into a solid hemisphere. What is the
ratio of the radius of the hemisphere to that of the cone?
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Answers
Given :- The radius of a solid sphere and a solid right circular cone are equal and the height of the cone is equal to the diameter of its base. They are melted and the combined matter is recast into a solid hemisphere. What is the ratio of the radius of the hemisphere to that of the cone ?
Solution :-
Let us assume that, radius of cone is r cm and Let us assume that , radius of hemisphere is R cm.
Than,
- Height of cone = Diameter of cone = 2r.
- Radius of sphere = r .
- Radius of hemisphere = R.
we know that,
- volume of cone = (1/3) * π * (radius)² * height.
- Volume of sphere = (4/3) * π * (radius)³ .
- volume of hemisphere = (2/3) * π * (Radius)³.
Than, after recast ,
→ Volume of cone + volume of sphere = volume of hemisphere
Putting all values we get :-
→ [(1/3) * π * (r)² * 2r] + [(4/3) * π * (r)³] = (2/3) * π * (R)³
→ (1/3)π[ 2r³ + 4r³ ] = (1/3)π[2R³]
→ 6r³ = 2R³
→ 3r³ = R³
→ (R³/r³) = (3/1)
cube root both sides
→ R/r = ³√3/1
→ R/r = (3)^1/3 / 1
→ R : r = (3)^(1/3) : 1 (Ans.)
Hence , Ratio of the radius of the hemisphere to that of the cone is (3)^(1/3) : 1 .
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