What is the least number of solid metallic spheres, each of 6 cm diameter, that should be melted and recast to form a solid metal cone whose height is 45 cm and diameter 12 cm ?

Given,

Diameter of metallic spheres = 6 cm

Radius of metallic spheres (r) = $\dfrac{6}{2}$ = 3 cm.

Height of cone (H) = 45 cm

Diameter of cone = 12 cm

Radius of cone (R) = $\dfrac{12}{2}$ = 6 cm.

Let no. of spheres melted to form a solid metal cone be n.

∴ n × Volume of each sphere = Volume of solid metal cone

$\Rightarrow n \times \dfrac{4}{3}πr^3 = \dfrac{1}{3}πR^2H \\[1em] \Rightarrow n = \dfrac{\dfrac{1}{3}πR^2H}{\dfrac{4}{3}πr^3} \\[1em] \Rightarrow n = \dfrac{πR^2H \times 3}{3 \times4 \times πr^3} \\[1em] \Rightarrow n = \dfrac{R^2H}{4r^3} \\[1em] \Rightarrow n = \dfrac{6^2 \times 45}{4 \times 3^3} \\[1em] \Rightarrow n = \dfrac{36 \times 45}{4 \times 27} \\[1em] \Rightarrow n = \dfrac{1620}{108} \\[1em] \Rightarrow n = 15.$

Hence, 15 solid metallic spheres need to be melted.