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 $= h$
Now, $\ell^2 = h^2 + r^2$
$\implies$ $h = \sqrt{\ell^2 - r^2} = \sqrt{13^2 - 5^2} = \sqrt{144} = 12 \ cm$
$\therefore$ Volume of cone $= \dfrac{1}{3} \pi r^2 h = \dfrac{1}{3} \times \pi \times 5^2 \times 12 = 100 \pi \ cm^3$
Radius of sphere $=R=1 \ cm$
$\therefore$ Volume of sphere $= \dfrac{4}{3} \pi R^3 = \dfrac{4}{3} \times \pi \times 1^3 = \dfrac{4}{3} \pi \ cm^3$
Let number of spheres $=N$
Since the cone is recast into a number of small spheres,
volume of cone $= N \times$ volume of 1 sphere
i.e. $100 \pi = N \times \dfrac{4}{3} \pi$
$\implies$ $N = \dfrac{100 \times 3}{4} = 75$
$\therefore$ Number of spheres recast = 75