A solid metallic cone of slant height $13 \; cm$ and radius $5 \; cm$ is melted and recast into solid spheres each of radius $1 \; cm$. Find the number of spheres recast.
Slant height of cone $= \ell = 13 \; cm$
Radius of cone $= r = 5 \; cm$
Let height of cone $= h \; cm$
$h = \sqrt{\ell^2 - r^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \; cm$
Volume of cone $= V_1 = \dfrac{1}{3}\pi r^2 h$
Let number of spheres made $= n$
Radius of each sphere $= R = 1 \; cm$
Volume of each sphere $= V_2 = \dfrac{4}{3} \pi R^3$
$\because \;$ The cone is melted and recast into $n$ spheres, we have,
$V_1 = n \times V_2$
i.e. $\;$ $\dfrac{1}{3} \pi r^2 h = n \times \dfrac{4}{3} \pi R^3$
i.e. $\;$ $n = \dfrac{r^2 h}{4 R^3}$
i.e. $\;$ $n = \dfrac{5^2 \times 12}{4 \times 1^3} = 75$
$\therefore \;$ Number of spheres made $= 75$