Question

A plane divides a cone into two parts of equal volume. If the plane is parellel to the base, in what ratio the height of the cone is divided by the plane
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Answer 1

Answer:

ANSWER

Let VAB be a cone of height h and base radius r. Suppose it is cut by a plane parallel to the base of the cone at point O

′

. Let O

′

A

′

=r

1

and VO

′

=h

1

Clearly,

∴

VO

′

VO

=

O

′

A

′

OA

⇒

h

1

h

=

r

1

r

It is given that

Volume of cone VA'B' = volume of the frustum ABB'A'

3

1

πr

1

2

h

1

=

3

1

π(r

2

+r

1

2

+rr

1

)(h−h

1

)

⇒r

1

2

h

1

=(r

2

+r

1

2

+rr

1

)(h−h

1

)

⇒1=

⎩

⎪

⎨

⎪

⎧

(

r

1

r

)

2

+1+

r

1

r

⎭

⎪

⎬

⎪

⎫

(

h

1

h

−1)⇒1=(

h

1

h

)

3

−1

3

⇒(

h

1

h

)

3

=2⇒

h

1

h

=2

1/3

Hence the ratio =

h−h

1

h

1

=

(

h

1

h

−1)

1

=

2

1/3

−1

1