Question

A solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm.  Find the number of cones so formed.

 

Answer 1

Radius of the solid metallic sphere, r = 10.5 cm
Radius of the cone, R = 3.5 cm
Height of the cone, H = 3 cm
Let the number of smaller cones formed be n. 
Volume of the metallic sphere,

\[V_s = \frac{4}{3}\pi \left( r \right)^3 = \frac{4}{3}\pi \left( 10 . 5 \right)^3\]

Volume of the cone, 

\[V_c = \frac{1}{3}\pi \left( R \right)^2 H = \frac{1}{3}\pi \left( 3 . 5 \right)^2 \times 3\]

Let the number of cones thus formed be n.

\[n \times \text { volume of the cone = volume of the sphere}\]

\[ \Rightarrow \frac{\text { volume of the sphere }( V_s )}{\text { volume of the cone }\left( V_c \right)} = n\]

\[ \Rightarrow \frac{\frac{4}{3}\pi \left( 10 . 5 \right)^3}{\frac{1}{3}\pi \left( 3 . 5 \right)^2 \times 3} = n\]

\[ \Rightarrow 126 = n\]

Hence, 126 cones are thus formed.