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Right triangle ABC has legs measuring 8 cm and 15 cm. The triangle is rotated about its hypotenuse. What is the number of cubic centimeters in the volume of the resulting solid? Express your answer in terms of π.

 Dec 2, 2020
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See the following image :

 

 

Let AC   = 8     BC  = 15      AB   =17

 

The area of this triangle   = (1/2) (8)(15)  =   60

 

The  altitude of  this triangle  can be found as

 

60    = (1/2)  17 * height

120 /17   = height  = CD  

 

This height will form the radius of two  cones

 

The height of the smaller  cone can be found as   sqrt  [ 8^2  - (120/17)^2  ] = 64/17  = 

The height of the larger cone can be found as   17 - 64/17  =  225/17

 

So   when   triangle  ACD  is rotated about hypotenuse  AB   its  volume is

(1/3) pi (120/17)^2 * (64/17)   =   [307200 / 4913 ] pi

 

And when  triangle BCD is rotated  about hypotenuse AB  its  volume id

(1/3)pi  ( 120/17)^2 (225/17)  = [1080000 / 4913 ]  pi

 

So...the total  volume  is  ( [  307200 +  1080000]  / 4913  )  pi  =   

 

[1387200 / 4913 ]  pi   cm^3 

 

 

cool cool cool

CPhill Dec 2, 2020

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