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 $=2 r$

$=2 \times 0.3$

$=0.6 \mathrm{~cm}$

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Updated on: 10-Oct-2022

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