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A hollow sphere of internal and external radii 6 cm and 8 cm respectively is melted and recast into small cones of base radius 2 cm and height 8 cm. Find the number of cones.

Mensuration

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Answer

Given,

External radii of hollow sphere (R) = 8 cm

Internal radii of hollow sphere (r) = 6 cm

Radius of cone (r1) = 2 cm

Height of cone (h) = 8 cm.

Let no. of cones formed after melting sphere be n.

∴ Volume of hollow sphere = n × Volume of cone

⇒43π(R3−r3)=n×13π(r1)2h⇒4(R3−r3)=n×(r1)2h⇒n=4(R3−r3)r12h⇒n=4[83−63]22×8=4×(512−216)32=4×29632=2968=37.\Rightarrow \dfrac{4}{3}π(R^3 - r^3) = n \times \dfrac{1}{3}π(r1)^2h \\[1em] \Rightarrow 4(R^3 - r^3) = n \times (r1)^2h \\[1em] \Rightarrow n = \dfrac{4(R^3 - r^3)}{r_1^2h} \\[1em] \Rightarrow n = \dfrac{4[8^3 - 6^3]}{2^2 \times 8}\\[1em] = \dfrac{4 \times (512 - 216)}{32} \\[1em] = \dfrac{4 \times 296}{32} \\[1em] = \dfrac{296}{8} \\[1em] = 37.

Hence, no. of cones formed = 37.

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