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Question

A cone is divided into two parts by drawing a plane through a point which divides its height in the ratio 1:2 starting from the vertex and the plane is parallel to the base. Compare the volume of the two parts

Solution
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Let the plane XY divide the cone ABC in the ratio AE:EO=12,
where AEO is the axis of the cone.
Let r2 and r1 be the radii of the circular section XY and the base BC of the cone respectively and let h1h and h1 be their heights.
Then,
h1h=32
=>h1=32h
And,
r1r2=h1h1h
=32h12h=3
r1=3r2
Volume of cone AXY
=13πr22(h1h)
=13πr22(32h1h)
=16πr22h
Volume of frustum XYBC
=13πh(r21+r22+r1r2)
=13πh(9r22+r22+3r22)
=13πh(13r22)
Volume of cone AXYVolume of frustum XYBC=16πr22h133πr22h2
=Volume of cone AXYVolume of frustum XYBC=126
The ratio between the volume of the cone AXY and the remaining portion BCYX is 1:26.

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