504 metallic cones, each of diameter 3.5 cm and height 3 cm, are melted and recast into a sphere. Find the diameter of sphere so formed.

Given,

For cone :

Diameter (d) = 3.5 cm

Radius (r) = $\dfrac{d}{2} = \dfrac{3.5}{2}$ = 1.75 cm

Height (h) = 3 cm

For sphere :

Let radius be R cm.

Since, 504 cones are melted and recasted into a sphere.

∴ 504 × Volume of each cone = Volume of sphere

$\Rightarrow 504 \times \dfrac{1}{3}πr^2h = \dfrac{4}{3}πR^3 \\[1em] \Rightarrow R^3 = \dfrac{504 \times 3 \times π \times r^2 \times h}{4 \times π \times 3} \\[1em] \Rightarrow R^3 = 126 \times (1.75)^2 \times 3 \\[1em] \Rightarrow R^3 = 1157.625 \\[1em] \Rightarrow R = \sqrt[3]{1157.625} \\[1em] \Rightarrow R = 10.5 \text{ cm}.$

Diameter = 2R = 2 × 10.5 = 21 cm.

Hence, diameter of sphere = 21 cm.