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.​

Answer 1

Answer:

Height of the cone = 28.4 cm

Step-by-step explanation:

Given:

• Radius of the hemisphere =  8 cm
• Radius of the cone = 6 cm

To Find:

• Height of the cone

Solution:

First we have to find the volume of the hemisphere.

Volume of a hemisphere is given by,

Volume of a hemisphere = 2/3 × π × r³

where r is the radius of the hemisphere

Substituting the data,

Volume of the hemisphere = 2/3 × 3.14 × 8³

⇒ 2/3 × 3.14 × 512

⇒ 1071.8 cm³

Hence volume of the hemisphere is 1071.8 cm³.

By given the hemisphere is melted to form a right circular cone.

Hence,

Volume of the cone = Volume of the hemisphere

Volume of a cone is given by,

Volume of  a cone = 1/3 × π × r² × h

Substituting the data,

1/3 × π × 6² × h = 1071.8

12 × 22/7 × h =1071.8

h = 1071.8/37.71

h = 28.4 cm

Therefore height of the cone is 28.4 cm.

Answer 2

$\begin{aligned}&\text { Volume of hemisphere }=\text { Volume of cone }\\\\&\therefore \frac{2}{3} \pi r^{3}=\frac{1}{3} \pi r^{2} h\\\\&\therefore \frac{2}{3} \pi 8^{3}=\frac{1}{3} \pi 6^{2} \mathrm{~h}\\\\&\therefore \mathrm{h}=\frac{1024}{36}=28.44\end{aligned}$

or

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