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Q.

Two solid cones A and B are placed in a cylindrical tube as shown in figure below. The ratio of their capacities is 2:1. What is the height of cones?

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a

14 cm and 7 cm

b

15 cm and 8 cm

c

19 cm and 6 cm

d

20 cm and 4 cm

answer is A.

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Detailed Solution

It is given that the ratio of the capacities of two solid cones A and B placed in a cylindrical tube are 2:1.
Finding the radius of the cone:
The diameter of the cone is 6cm.

r = \frac{d}{2} = \frac{6}{2} = 3 cm

The volume of both the cones.
Let

h 1

be the height of the first cone. Then the height of the second cone will be

21 − h 1

. As the ratio of volume is 2:1, volume of the first cone is 2V and the volume of the second cone is V. So, for the first cone,
The volume of the cone of radius r and height

h is, 13 π r 2 h

.

2 V = 13 π r 2 h 2 V = 13 π 32 h 1 V = 96 π h 1 V = 32 π h 1 … … 1

For second cone,

V = 13 π r 2 h V = 13 π 32 21 − h 1 V = 3 π 21 − h 1 … … 2

Comparing the equations (1) and (2) to determine the value of

h 1

.

32 π h 1 = 3 π 21 − h 1 h 1 = 2 21 − h 1 3 h 1 = 42 h 1 = 14 c m

Therefore, the height of the second cone is,

21 − h 1 = 21 − 14 h 1 = 7 c m

Hence 1 is the correct option.

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