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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?

 Sep 25, 2023
 #1
avatar+2 
0

The formula for the volume of a sphere is given by:

Volume of Sphere = (4/3)πr^3

                              = (4/3)π(3^3) = 36π cubic units.

The formula for the volume of a cone is given by:

Volume of Cone = (1/3)πr^2h

                           = (1/3)π(3^2)(3) = 27π cubic units

Ratio = (Volume of Sphere) / (Volume of Cone)

         = (36π) / (27π) = 4/3

 Sep 25, 2023
 #2
avatar+128732 
+1

Through the use of similar triangles, we can find the radius, r,  of the  sphere thusly :

 

 

Let one right triangle CDB be composed  of  two  legs.....DB =  the base radius  of the  cone = 3 and  the other DC = the height of the cone = 3

 

A simiar  second  right riangle CFE  can be  formed   (see the illustration) with legs EF =  r and FC =  r

The hypotenuse of this triangle = CE =   sqrt [ r^2 + (r^2) ] = r sqrt 2

 

And we  can  find the  radius of the  sphere thusly :

 

CD = 3

 

ED + CE  =  3

 

r + r sqrt 2  = 3

 

r ( 1 + sqrt 2)  = 3

 

r = 3 / ( 1 + sqrt 2)

 

Volume of sphere to volume of the cone  =

 

(4/3) pi ( 3 / (1+sqrt 2) )^3                        4

_____________________  =      ____________ ≈   .284

(1/3) pi ( 3)^2 * 3                           (1 + sqrt 2)^3

 

cool cool cool

 Sep 25, 2023

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