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What will be the height of the cone, if a metallic spherical shell of internal and external diameters 4 cm and 8 cm respectively, is melted and recast into the form of a cone of base diameter 8cm?

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12 cm

14 cm

15 cm

8 cm

answer is B.

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Detailed Solution

Given,
A metallic spherical shell of internal and external diameters 4 cm and 8 cm respectively.
We have to find the height of the cone.
We know that,

d i a m e t e r = 2 × r a d i u s

Constructing the required figure.
Question Image

Calculating the Internal radius of a spherical shell,

r 1 = 42 c m

= 2 c m

Calculating the external radius of the shell,

r 2 = 82 c m

= 4 cm

Note that the volume of the spherical shell,

= 43 π (external radius) 3 - (internal radius) 3

Note that volume of the spherical shell

= 43 π r 23 − r 13

= 43 π 43 − 23

= 43 π (64 − 8)

= 2243 π cm 3

Let the height of the cone

= h cm

and it is given that the diameter of the base of cone

= 8 cm

So, the radius of the base of the cone

= 82

= 4 cm

Note that the volume of a cone,

= 13 × π × (radius) 2 × (h e i g h t)

Solving according to the question,
Volume of cone = Volume of a spherical shell.

⇒ 13 π (4) 2 h = 2243 π

⇒ h = 22416 = 14 c m

So, the height of the cone is

14 c m .

Therefore, option 2 is correct.

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