+0  
 
0
1
1
avatar+372 

Triangle ABC is rotated completely around side BC, sweeping out a solid in space. Find the volume of the solid.

AB = 8, AC = 12, CB = 8

 Jun 1, 2024
 #1
avatar+129487 
+1

This will be the difference in the volumes formed by 2 cones....see the foklowing

 

 

To find the radius of the cones, we need to find the   area of the triangle  (using Heron's formula)

 

Area =  sqrt  [ 14 *  6 * 6 *  2]  = 12sqrt 7

 

To find the  radius

 

12sqrt 7  =  (1/2) BC * height

 

12sqrt 7  = (1/2) (8) * height

 

3sqrt 7  = height  = radius  of both cones =  sqrt [ 63]  = AD

 

To find the height of one of the cones we have

sqrt [ 8^2 - 63 ] =  sqrt (1)  = 1

To find the height of the other cone we have 

sqrt [ 12^2 - 63 ] =sqrt (81)  = 9

 

Total volume  of rotated  triangle ABC =  Volume formed by rotation of triangle ADC - volume formed by rotation of triangle ABD  =

 

(1/3) pi (63) (9 - 1)  =  168 pi ≈ 527.79

 

 

cool cool cool

 Jun 1, 2024

1 Online Users