b
(b) Apparent depth \(h' = \frac{h}{{_{air}{\mu _{liquid}}}}\)

==> \(\frac{{dh'}}{{dt}} = \frac{1}{{_a{\mu _w}}} = \frac{1}{{_a{\mu _w}}}\frac{{dh}}{{dt}}\)

==>\(x = \frac{1}{{_a{\mu _w}}}\frac{{dh}}{{dt}}\)==> \(\frac{{dh}}{{dt}} = {\,_a}{\mu _w}\,x\)

Now volume of water \(V = \pi {R^2}h\)

==> \(\frac{{dV}}{{dt}}\) \( = \pi {R^2}\frac{{dh}}{{dt}}\) \( = \pi {R^2}.{\,_a}{\mu _w}\,x\)

\( = {\,_a}{\mu _w}\pi {R^2}x\) \( = \frac{{{\mu _w}}}{{{\mu _a}}}\pi {R^2}x\)\( = \left( {\frac{{{n_2}}}{{{n_1}}}} \right)\,\pi {R^2}x\)