Explanation

(b) Longitudinal strain $\alpha = \ \frac{l_{2} - l_{1}}{l_{1}} = 10^{- 3}$
$\frac{l_{2}}{l_{1}} = 1.001$
Poisson’s ratio, $\sigma = \ \frac{lateral\ strain}{longitudinal\ strai} = \frac{\beta}{\alpha}\ $
Or $\beta = \ \sigma\alpha = 0.1\ \times 10^{- 3} = 10^{- 4} = \frac{r_{1} - r_{2}}{r_{1}}$
Or $\frac{r_{2}}{r_{1}} = 1 - 10^{- 4} = 0.9999$
% increase in volume = $\left( \frac{V_{2} - V_{1}}{V_{1}} \right) \times 100$
=$\left( \frac{\pi r_{2}^{2}l_{2} - \pi_{1}^{2}l_{1}}{\pi r_{1}^{2}l_{1}} \right) \times 100 = \left( \frac{r_{2}^{2}l_{2}}{r_{1}^{2}l_{1}} - 1 \right) \times 100$
= [$(0.9999)^{2} \times 1.001 - 1\rbrack\ \times 100 = 0.08\%$