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A sphere is inscribed in a cone with height $3$ and base radius $3$. What is the ratio of the volume of the sphere to the volume of the cone?

 Mar 29, 2024
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Note that  triangle ADO  is similar to triangle AEC

AO / DO  =  AC / EC

 

AO  = 3 - r

DO  = r

AC =  sqrt  [ 3^2 + 3^2 ] =  3sqrt [2]

EC  =  3

 

(3 - r)  / r =  3sqrt (2 )  / 3

  3 - r  = r *sqrt (2)

3 = r [ 1 + sqrt 2]

r = 3 / [1 + sqrt 2] ≈ 1.24

 

Volume of sphere   = (4/3)(pi)(1.24)^3  ≈  2.54 pi  =  A

Volume of cone = (1/3) pi (3^2) * 3  = 9pi  =  B

 

A / B  ≈  2.54 / 9  =   127 / 450

 

cool cool cool

 Mar 29, 2024

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