State true or false.The formula for the volume of the frustum of a cone is

1
3

πh(
r
1
2

+
r
2
2

+
r
1

r
2

)
, given to you in the figure below is, using the symbols as explained.

Question

State true or false.The formula for the volume of the frustum of a cone is

1
    3
   
   πh(
    r
    1
    2
   
   +
    r
    2
    2
   
   +
    r
    1

r
    2
   
   )
  , given to you in the figure below is, using the symbols as explained.

Accepted Answer

Given is a frustum of a cone with h as height, l as the slant height,

,
    r
    1

and

r
    2

radii of the ends where

r
    1
   
   ,
  >

r
    2

Extend the side BC and AD of the frustum of the cone to meet at O.From the figure:The frustum of a cone can be viewed as a difference of two right circular cones OAB and OCD.Let

h
    1

and

l
    1

be the height and slant height of cone OAB and

h
    2

and

l
    2

be the height and slant height of cone OCD respectively.

ΔAPO

andΔDQO
  As both cones are right angled,

ΔAPO=ΔDQO=
      90
      0

∠AOP=∠DOQ(Commonangle)

Therefore,ΔAPO~ΔDQO
      
       A.Acriterion

AP
      
       DQ
     
     =
      
       AO
      
       DO
     
     =
      
       OP
      
       OQ
     
     (byCPCT)

Put the values:

r
        1

r
        2

=

l
        1

l
        2

=

h
        1

h
        2

⇒

r
        1

r
        2

=

h
        1

h
        2

⇒

r
        2

r
        1

=

h
        2

h
        1

Subtract 1 from both sides again :Volume of frustum of cone = Volume of cone OAB - Volume of cone OCD:

⇒Volume=
      1
      3
     
     π
      r
      1
      2

h
      1
     
     −
      1
      3
     
     π
      r
      2
      2

h
      2

⇒Volume=
      1
      3
     
     π

r
        1
        2

h
        1
       
       −
        r
        2
        2

h
        2

⇒Volume=
      1
      3
     
     π

r
        1
        2
       
       ×
        
         h
          r
          1

r
          1
         
         −
          r
          2

−
        r
        2
        2
       
       ×
        
         h
          r
          2

r
          1
         
         −
          r
          2

Simplify the equation:Hence, volume of the frustum of a cone

=
    1
    3
   
   πh(
    r
    1
    2
   
   +
    r
    2
    2
   
   +
    r
    1

r
    2
   
   )
  Therefore, the given statement is true.Therefore, the correct answer is option (1).

State true or false.

The formula for the volume of the frustum of a cone is $13 π h (r 12 + r 22 + r 1 r 2)$ , given to you in the figure below is, using the symbols as explained.

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State true or false.The formula for the volume of the frustum of a cone is 1 3 πh( r 1 2 + r 2 2 + r 1 r 2 ) , given to you in the figure below is, using the symbols as explained.

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State true or false.

The formula for the volume of the frustum of a cone is $13 π h (r 12 + r 22 + r 1 r 2)$ , given to you in the figure below is, using the symbols as explained.