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Mathematics

A solid metallic hemisphere of diameter 28 cm is melted and recast into a number of identical solid cones, each of diameter 14 cm and height 8 cm. Find the number of cones so formed.

Mensuration

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Answer

Given,

Diameter of metallic hemisphere = 28 cm

Radius of metallic hemisphere (R) = 282\dfrac{28}{2} = 14 cm.

Diameter of cone = 14 cm

Radius of cone (r) = 142\dfrac{14}{2} = 7 cm

Height (h) = 8 cm.

Let no. of cones formed be n.

Solid metallic hemisphere is melted and recasted into cones.

∴ Volume of metallic hemisphere = n × Volume of each cone

⇒23πR3=n×13πr2h⇒2R3=n×r2h⇒n=2R3r2h⇒n=2×14372×8⇒n=2×274449×8⇒n=5488392=14.\Rightarrow \dfrac{2}{3}πR^3 = n \times \dfrac{1}{3}πr^2h \\[1em] \Rightarrow 2R^3 = n \times r^2h \\[1em] \Rightarrow n = \dfrac{2R^3}{r^2h} \\[1em] \Rightarrow n = \dfrac{2 \times 14^3}{7^2 \times 8} \\[1em] \Rightarrow n = \dfrac{2 \times 2744}{49 \times 8} \\[1em] \Rightarrow n = \dfrac{5488}{392} = 14.

Hence, no. of cones formed = 14.

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