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If a sphere of radius ' $a$ ' unit is immersed completely in water contained in a right circular cone of semi-vertical angle $30 °$ and water is drained off from the cone till its surface touches the sphere, then what is the volume of water remaining in the cone?

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3 π 5 a 2

3 π a 2

23 π a 3

53 π a 3

answer is D.

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Detailed Solution

Given that a sphere of radius '

a

' and a right circular cone of semi-vertical angle

30 °

We have the following formulae,’
Volume of cone =

13 π r 2 h

Volume of sphere =

43 π r 3

We have,
Radius of sphere

= a

Δ A O B

sin 30 ° = O B O A

Height of cone will be,

∴ AC = O A + O C

= 2 a + a = 3 a

Δ A C D

tan 30 ° = C D A C

C D = A C × tan 30 ° = 3 a 3 3 a

Radius of cone

= 3 a

Volume of water remaining in cone = volume of cone - volume of sphere

= 13 π r 2 h − 43 π r 3

= π a 3 3 − 43

= 53 π a 3

Thus the volume of the remaining cone is

53 π a 3

.
Therefore the correct option is 4.

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