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

Mensuration

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Answer

Given,

Radius of cone (r) = 2 cm,

Height of cone (h) = 3 cm,

Radius of sphere (R) = 6 cm.

Let n cones be melted to recast into a solid sphere.

∴ n × Volume of each cone = Volume of sphere

⇒n×13πr2h=43πR3⇒n=43πR313πr2h⇒n=4πR3×3πr2h×3⇒n=4R3r2h⇒n=4×6322×3⇒n=86412⇒n=72.\Rightarrow n \times \dfrac{1}{3}πr^2h = \dfrac{4}{3}πR^3 \\[1em] \Rightarrow n = \dfrac{\dfrac{4}{3}πR^3}{\dfrac{1}{3}πr^2h} \\[1em] \Rightarrow n = \dfrac{4πR^3 \times 3}{πr^2h \times 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{864}{12} \\[1em] \Rightarrow n = 72.

Hence, 72 cones need to be melted to recast into a solid sphere.

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