A certain number of metallic cones each of radius 2 cm and height 3 cm are melted and recast in a solid sphere of radius 6 cm. Find the number of cones.

Radius of each cone (r) = 2 cm

Height of cone (h) = 3 cm.

Let the number of cones required to be recast into a solid sphere of radius (R) = 6 cm be n.

n × Volume of each cone = Volume of sphere.

$\therefore n \times \dfrac{1}{3}πr^2h = \dfrac{4}{3}πR^3$

On dividing both sides by π and multiplying by 3,

$\Rightarrow n \times r^2h = 4R^3 \\[1em] \Rightarrow n = \dfrac{4R^3}{r^2h} \\[1em] \Rightarrow n = \dfrac{4 \times (6)^3}{(2)^2 \times 3} \\[1em] \Rightarrow n = \dfrac{4 \times 216}{4 \times 3} \\[1em] \Rightarrow n = \dfrac{864}{12} \\[1em] \Rightarrow n = 72.$

Hence, the number of cones required to make a solid sphere of radius 4 cm are 72.