 # The diameter of a copper sphere is 18 cm. It is melted and drawn into a long wire of uniform cross-section. `
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.

Solution:

Diameter of sphere $=18 \mathrm{~cm}$

Radius of the sphere $=9 \mathrm{~cm}$

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

Length of wire $=108 \mathrm{~m}=10800 \mathrm{~cm}$

Let radius of the wire be $r \mathrm{~cm} .$

Volume of the wire $=\pi r^{2} l=\pi \times r^{2} \times 10800 \mathrm{~cm}^{3}$

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

$\pi \times r^{2} \times 10800=\frac{4}{3} \pi \times 9 \times 9 \times 9$

$\Rightarrow r^{2}=\frac{4 \times 9 \times 9 \times 9}{3 \times 10800}=\frac{4 \times 9 \times 9}{3 \times 4 \times 3}=0.09$

$\Rightarrow r=\sqrt{0.09}=0.3 \mathrm{~cm}$

Thus, $d=2 r=2 \times 0.3 \mathrm{~cm}=0.6 \mathrm{~cm}$

The diameter of the wire is 0.6 cm.