$\frac{\text { Lateral strain }}{\text { Longitudinal strain }}=\eta=0.5$

$\frac{-\Delta r / r}{\Delta l / l}=\frac{1}{2}$

$\frac{-2 \Delta r}{r}=\frac{\Delta l}{l}$

Magnitute wise both are equal but sign's would be different as both quantities cannot increase

Now volume $\propto$ area $\times$ length $v \propto r^2 \cdot L$

$\frac{\Delta V}{V}=\frac{2 \Delta r}{r}+\frac{\Delta L}{L}$

Substituting value of $\frac{\Delta L}{L}$

$\frac{\Delta V}{V}=0$

$\therefore$ No change in volume.