The surface area of a solid metallic sphere is $2464 \; cm^2$. It is melted and recast into solid right circular cones of radius $3.5 \; cm$ and height $7 \; cm$. Calculate the radius of the sphere and the number of cones recast. Take $\pi = \dfrac{22}{7}$.
Let the radius of the sphere $= R$
Surface area of sphere $= 4 \pi R^2 = 2464$
i.e. $\;$ $R^2 = \dfrac{2464}{4 \pi} = \dfrac{2464 \times 7}{4 \times 22} = 196$ $\implies$ $R = \sqrt{196} = 14 \; cm$
$\therefore \;$ Radius of sphere $= R = 14 \; cm$
Radius of cone $= r = 3.5 \; cm$
Height of cone $= h = 7 \; cm$
Let the number of cones recast $= n$
Since the sphere is melted and recast into $n$ number of cones,
Volume of sphere $= n \times$ volume of cone
i.e. $\;$ $\dfrac{4}{3} \pi R^3 = n \times \dfrac{1}{3} \pi r^2 h$
i.e. $\;$ $n = \dfrac{4 R^3}{r^2 h} = \dfrac{4 \times 14^3}{3.5^2 \times 7} = 128$
$\therefore \;$ Number of cones recast $= 128$