Find the equivalent thermal conductivity of the system.

$A$ cylindrical copper rod of length $2 \,m$ and cross-sectional area $2 \,cm^2$ is insulated at its curved surface. One end of the rod is maintained in a steam chamber at $100^{\circ} C$ and the other is maintained in ice at $0^{\circ} C$. The thermal conductivity of copper is $386 \,Js^{-1} \,m^{-1} {}^{\circ} C^{-1}$. Find the temperature at a point which is at a distance of $120 \,cm$ from the colder end. (in $^{\circ} C$)

Two plates of equal area are placed in series. The ratios of their thicknesses and thermal conductivities are both $2:3$. The temperature of the outer surface of one plate is $100 ^\circ C$ and that of the other is $0 ^\circ C$. The temperature of the common surface is ....... $^\circ C$.

An ice cube of dimensions $60\,cm \times 50\,cm \times 20\,cm$ is placed in an insulation box of wall thickness $1\,cm$. The box keeping the ice cube at $0^{\circ}C$ temperature is brought to a room of temperature $40^{\circ}C$. The rate of melting of ice is approximately. (Latent heat of fusion of ice is $3.4 \times 10^{5}\,J\,kg^{-1}$ and thermal conductivity of insulation wall is $0.05\,W\,m^{-1\circ}C^{-1}$)

Two layers of a wall,$A$ and $B$,are made of different materials. Both layers have the same thickness. For layer $A$,the thermal conductivity is $K_A = 3 K_B$. The total temperature difference across the wall is $20^{\circ}C$. Find the temperature difference across layer $A$ in $^{\circ}C$.

Two spheres of different materials,one with double the radius and one-fourth wall thickness of the other,are filled with ice. If the time taken for complete melting of ice in the large radius sphere is $25 \ minutes$ and that for the smaller one is $16 \ minutes$,the ratio of thermal conductivities of the materials of the larger sphere to the smaller sphere is: