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Question

If a cone is cut into two parts by a horizontal plane passing through the mid-point of its axis, then the ratio of the volumes of the upper part of the cone and the entire cone is:


A

1:2

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B

1:4

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C

1:6

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D

1:8

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Solution

The correct option is D

1:8



Let the radius and height of the given cone = R and H.
Let the radius and height of the upper part of the cone = r and h.
Volume of the given cone=13πR2H
From the figure:
AQD ~APC
AQAP=QDPC=ADAC
hH=rR
H2H=rR [h=H2]
rR=12
Ratio of the volume between the upper part of the cone and the entire cone is

13πr2h:13πR2H=r2hR2H=(12)2×H2H=18
Hence the ratio is 1:8.


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