a
(a) Rate of flow of heat is given by \(\frac{{dQ}}{{dt}} = \frac{{\Delta \theta }}{{l/KA}}\)also \(\frac{{dQ}}{{dt}} = L\frac{{dm}}{{dt}}\) (where \(L\) = Latent heat)

==> \(\frac{{dm}}{{dt}} = \frac{{KA}}{l}\,\left( {\frac{{\Delta \theta }}{L}} \right)\). Let the desire point is at a distance \(  x\) from water at \(100°C\) . ( Rate of ice melting = Rate at which steam is being produced

==> \({\left( {\frac{{dm}}{{dt}}} \right)_{Steam}} = {\left( {\frac{{dm}}{{dt}}} \right)_{Ice}}\)

==> \({\left( {\frac{{\Delta \theta }}{{Ll}}} \right)_{Steam}} = {\left( {\frac{{\Delta \theta }}{{Ll}}} \right)_{Ice}}\)

==> \(\frac{{(200 - 100)}}{{540 \times x}} = \frac{{(200 - 0)}}{{80\,(3.1 - x)}}\)

==> \(x = 0.4 m = 40 cm\)