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The diameter of a copper sphere is $18\ cm$. The sphere is melted and is drawn into a long wire of uniform circular cross-section. If the length of the wire is $108\ m$, find its diameter.
Given:
The diameter of a copper sphere is $18\ cm$. The sphere is melted and is drawn into a long wire of uniform circular cross-section.
The length of the wire is $108\ m$.
To do:
We have to find its diameter.
Solution:
Diameter of the copper sphere $= 18\ cm$
This implies,
Radius of the sphere $=\frac{18}{2}$
$=9 \mathrm{~cm}$
Volume of the sphere $=\frac{4}{3} \pi r^{3}$
$=\frac{4}{3} \times \pi \times(9)^{3}$
$=972 \pi \mathrm{cm}^{3}$
Length of the wire $(h)=108 \mathrm{~m}$
$=108 \times 100 \mathrm{~cm}$
Volume of the wire $=972 \pi \mathrm{cm}^{3}$
Therefore,
Radius of the wire $=\sqrt{\frac{\text { Volume }}{\pi h}}$
$=\sqrt{\frac{972 \pi}{\pi \times 108 \times 100}} \mathrm{~cm}$
$=\sqrt{\frac{9}{100}} \mathrm{~cm}$
$=\frac{3}{10}$
$=0.3 \mathrm{~cm}$
Diameter of the wire $=2 r$
$=2 \times 0.3$
$=0.6 \mathrm{~cm}$