\(\frac{\text { Lateral strain }}{\text { Longitudinal strain }}=\eta=0.5\)

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

\(\frac{-2 \Delta r}{r}=\frac{\Delta l}{l}\)

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.