Math, asked by ande29071999, 5 months ago


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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1961 10
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Answers

Answered by RvChaudharY50
24

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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