Question

The diameter of a copper sphere is 18 cm. It is melted and drawn into a long wire of uniform cross-section. If the length of the wire is 108 m, find its diameter.

Answer 1

Diameter of sphere = 18 cm
Radius of the sphere = 9 cm

Volume of sphere$=\frac{4}{3}{\mathrm{\pi r}}^3=\frac{4}{3}\mathrm{\pi }\times 9\times 9\times 9{\mathrm{cm}}^3$

Length of wire = 108 m = 10800 cm
Let radius of the wire be r cm.
Volume of the wire$=\pi r^{2}l=\mathrm{\pi }\times r^2\times 10800{\mathrm{cm}}^3$

The volume of the sphere and the wire are the same.
Therefore,

$$\begin{gathered} \mathrm{\pi }\times r^2\times 10800=\frac{4}{3}\mathrm{\pi }\times 9\times 9\times 9 \\ \Rightarrow r^2=\frac{4\times 9\times 9\times 9}{3\times 10800}=\frac{4\times 9\times 9}{3\times 4\times 3}=0.09 \\ \Rightarrow r=\sqrt{0.09}=0.3\mathrm{cm} \\ \mathrm{Thus},d=2r=2\times 0.3\mathrm{cm}=0.6\mathrm{cm} \end{gathered}$$

The diameter of the wire is 0.6 cm.