Question

A solid metallic hemisphere of radius 8 cm is melted and recasted into a right circular cone of base radius 6 cm. Determine the height of the cone

NCERT Class X
Mathematics - Exemplar Problems

Chapter _SURFACE AREAS AND VOLUMES

Answer 1

Hi,

Radius of the hemisphere = 8 cm

R = 8 cm

Volume of the hemisphere

= ( V ) = ( 2 / 3) × pi × R^3 ---------(1)

Radius of the right circular cone = r

r = 6 cm

Let the height of the cone = h cm

Volume of the right circular cone

= v = ( 1/ 3 ) × pi × r ^2 × h -------(2)

According to the problem,

The solid metallic sphere is melted

and recasted into a right circular

cone.

Therefore,

Volume of the hemisphere and

volume of the right circular cone are

equal.

( 2 ) = ( 1 )

( 1/ 3) × pi × r ^2 × h = ( 2/3 ) × pi × R ^3

After cancellation

h = 2 × ( R ^3 / r ^2 )

Substitute R and r values

h = 2 × ( 8 × 8 × 8 )/ ( 6 × 6 )

h = ( 4 × 8 × 8 ) / ( 3 × 3 )

h = 256 / 9

h = 28. 44 cm

Height of the cone = h = 28.44 cm

I hope this will useful to you.

******

Answer 2

For hemisphere,

Radius, r = 8 cm

As we know,

Volume of hemisphere=1/3×pie×r2×h

Where, r = radius of hemisphere

Volume of given hemisphere

For cone that is recast from hemisphere,

Base radius, r = 6 cm

We know that,

volume of cone

Where, r is base radius and h is the height of the cone.

Now,

Volume of cone

As the volume remains same, when a body is reformed to another body

Volume of cylinder = Volume of cone

12πh = 1024π /3

h = 28.44 cm

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