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Mathematics

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 ?

Mensuration

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Answer

Given,

Diameter of metallic spheres = 6 cm

Radius of metallic spheres (r) = 62\dfrac{6}{2} = 3 cm.

Height of cone (H) = 45 cm

Diameter of cone = 12 cm

Radius of cone (R) = 122\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

⇒n×43πr3=13πR2H⇒n=13πR2H43πr3⇒n=πR2H×33×4×πr3⇒n=R2H4r3⇒n=62×454×33⇒n=36×454×27⇒n=1620108⇒n=15.\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.

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