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There is a cone.The slant height of cone is 6√3 and radius is 3√3. There is a sphere inscribed in the cone(touching its base and curved surface). Find the radius of sphere.

 

 Dec 22, 2019
 #1
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We can solve this through the method of  cross-section

 

See the following image  :

 

 

6sqrt (3)     = sqrt (108)      and 3sqrt (3)  = sqrt (27)

 

Note that triangle ABC is right......and we can find AB   as   sqrt  (  108 - 27)  =  sqrt (81)  =  9

 

Draw BE  perpendicular to AC

 

And triangles   CEB  and CBA  are similar....so

 

BC/ AC  = BE / AB

 

3sqrt (3) / 6sqrt (3) = BE / 9

 

3/ 6  =  BE / 9

 

1/2  = BE/ 9

 

BE  = 9/2  = 4.5

 

And triangle AFG  is similar to triangle  ABE

 

So

 

AF / FG =  AB / BE

 

9 - R                 9    

____     =     ______     cross-multiply

   R                 4.5

 

4.5 (9 - R)  =  9R

 

40.5  - 4.5R = 9R

 

40.5  =  13.5R

 

R  = 3     = the radius of the sphere

 

 

cool cool cool

 Dec 22, 2019

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