A metallic sphere of radius 10.5 cm is melted and then recast into small cones, each of radius 3.5 cm and height 3 cm. Find the number of cones thus obtained.

Radius of sphere (r) = 10.5 cm

Let the number of cones formed by recasting metallic sphere = n.

Radius of cone (R) = 3.5 cm

Height (h) = 3 cm.

Volume of sphere = n × Volume of each cone.

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

Dividing both sides by π and multiplying by 3,

$\Rightarrow 4r^3 = n \times R^2h \\[1em] \Rightarrow 4 \times (10.5)^3 = n \times (3.5)^2 \times 3 \\[1em] \Rightarrow n = \dfrac{4 \times 1157.625}{12.25 \times 3} \\[1em] \Rightarrow n = \dfrac{4630.5}{36.75} \\[1em] \Rightarrow n = 126.$

Hence, the number of cones obtained from metallic sphere are 126.