+782 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
+129771 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
