A copper rod of diameter $ 1 \mathrm{~cm} $ and length $ 8 \mathrm{~cm} $ is drawn into a wire of length $ 18 \mathrm{~m} $ of uniform thickness. Find the thickness of the wire.

A copper rod of diameter \( 1 \mathrm{~cm} \) and length \( 8 \mathrm{~cm} \) is drawn into a wire of length \( 18 \mathrm{~m} \) of uniform thickness.

To do:

We have to find the thickness of the wire.

Solution:

Diameter of the rod $=1 \mathrm{~cm}$

This implies,

Radius of the cone $\mathrm{R}=\frac{1}{2} \mathrm{~cm}$

Height of the cone $H=8 \mathrm{~cm}$

Therefore,

Volume of the cone $=\pi \mathrm{R}^{2} H$

$=\pi \times(\frac{1}{2})^{2} \times 8 \mathrm{~cm}^{3}$

$=\pi \times \frac{1}{4} \times 8$

$=2 \pi \mathrm{cm}^{3}$

Length of the wire drawn $h=18 \mathrm{~m}$

$=1800 \mathrm{~cm}$

Let the radius of the wire be $r$.

Therefore,

Volume of the wire $=\pi r^{2} h$

$=\pi r^{2} \times 1800$

$\Rightarrow 1800 \pi r^{2}=2 \pi$

$\Rightarrow r^{2}=\frac{2 \times \pi}{1800 \times \pi}$

$\Rightarrow r^{2}=\frac{1}{900}$

$\Rightarrow r^{2}=(\frac{1}{30})^{2}$

$\Rightarrow r=\frac{1}{30} \mathrm{~cm}$

$\Rightarrow r=\frac{100}{30} \mathrm{~mm}$

$\Rightarrow r=\frac{10}{3} \mathrm{~mm}$

This implies,

Diameter of the wire $=2 r$

$=2 \times \frac{10}{3}$

$=\frac{20}{3}$

$=6.67 \mathrm{~mm}$

The thickness of the wire is $6.67\ mm$.