Question

The radius of a wire is decreased to one-third. If volume remains the same, the length will be increased __________.

Answer 1

Let r units and l units be the radius and length of the wire, respectively.

∴ Original volume of the wire = $\mathrm{\pi }$r²l cu. units

When the radius of the wire is decreased to one-third, then the new radius of the wire, R = $\frac{r}{3}$ units.

Suppose the new length of the wire is L units.

∴ New volume of the wire = $\mathrm{\pi }{\left(\frac{r}{3}\right)}^{2}L$ cu. units

It is given that, the volume of the wire remains the same.

$\therefore \mathrm{\pi }{\left(\frac{r}{3}\right)}^{2}L=\mathrm{\pi }{r}^{2}l$

$\Rightarrow \frac{r^2}{9}\times L=r^2\times l$

$\Rightarrow L=9l$

Thus, the length of the wire will be increased 9 times.

The radius of a wire is decreased to one-third. If volume remains the same, the length will be increased ___9 times___.