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प्रश्न
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.
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उत्तर
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.
