A solid cylinder of radius $7 \; cm$ and height $14 \; cm$ is melted and recast into solid spheres of radius $3.5 \; cm$. Find the number of spheres formed.
Radius of cylinder $= r = 7 \; cm$
Height of cylinder $= h = 14 \; cm$
Volume of cylinder $= V_c = \pi r^2 h = \pi \times 7^2 \times 14 \; cm^3$
Radius of sphere $= R = 3.5 \; cm$
Volume of sphere $= V_s = \dfrac{4}{3} \pi R^3 = \dfrac{4}{3} \times \pi \times 3.5^3 \; cm^3$
Let the number of spheres formed $= n$
$\because \;$ the cylinder is melted and recast into a number of spheres,
$\text{Volume of cylinder} = n \times \text{Volume of sphere}$
$\begin{aligned}
\therefore \; n & = \dfrac{\text{Volume of cylinder}}{\text{Volume of sphere}} \\\\
& = \dfrac{\pi \times 7^2 \times 14}{\dfrac{4}{3} \times \pi \times 3.5^3} \\\\
& = \dfrac{3 \times 7^2 \times 14}{4 \times 3.5^3} = 12
\end{aligned}$
$\therefore \;$ Number of spheres formed $= 12$