State true or false.A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. Then the ratio of the volume of the smaller cone to the whole cone is 1: 8.

Question

State true or false.A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. Then the ratio of the volume of the  smaller cone to the whole cone is 1: 8.

Accepted Answer

We have to check if the ratio of the volume of the  smaller cone to the whole cone is 1: 8.Drawing the required figure.Suppose the radii of the smaller and given cones are r and R respectively and their height are h and H respectively. Therefore,

H=2h          ……(1)
  Note that, In

ΔAQD and ΔAPC,

∠QAD=∠PAC

∠AQD=∠APC=
        
         90
        °

And, by AA criteria,

ΔAQD∼ΔAPC

⇒
        
         AQ
        
         AP
       
       =
        
         QD
        
         PC

⇒
        h
        H
       
       =
        r
        R

Solving further,

⇒
        h
        
         2h
       
       =
        r
        R

⇒
        1
        2
       
       =
        r
        R

⇒R=2r           ……(2)

Volume of the cone = 
    1
    3
   
   π
    r
    3
   
   h
   , where r is the radius, h is the height.Now the ratio of the smaller cone to the whole cone,

= 
    
      Volume of the smaller cone
    
      Volume of the whole cone

=

1
        3
       
       π
        r
        2
       
       h

1
        3
       
       π
        R
        2
       
       H

=

r
        R

2
   
   ×

h
      H

Solving further,

=

1
        2

2
   
   ×

1
      2

=
    1
    8

=1:8
   So, the ratio of the smaller cone to the whole cone is

1:8
   and thus the statement is true.Therefore, option 1 is correct.

State true or false.

A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. Then the ratio of the volume of the smaller cone to the whole cone is 1: 8.

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