Why do some cooking vessels have a copper coating at the bottom?

In the figure shown,$AB$ is a rod of length $30 \ cm$,area of cross-section $1 \ cm^2$ and thermal conductivity $336 \ SI$ units. The ends $A$ and $B$ are at constant temperatures $20^{\circ} C$ and $40^{\circ} C$ respectively. $A$ point $C$ of the rod is connected to ice at $0^{\circ} C$ in a thermally insulated box $D$ through a highly conducting wire of negligible heat capacity. The rate at which ice melts in the box is $\left(L_{ice}=80 \ cal \ g^{-1}\right)$.

$A$ $5 \ cm$ thick ice block is present on the surface of water in a lake. The temperature of the air is $-10^{\circ}C$. How much time will it take to double the thickness of the block? (Given: $L = 80 \ cal/g$,$K_{ice} = 0.004 \ cal/s \cdot cm \cdot ^{\circ}C$,$\rho_{ice} = 0.92 \ g/cm^3$)

On heating one end of a rod,the temperature of the whole rod will be uniform when

One end of a thermally insulated rod is kept at a temperature $T_1$ and the other at $T_2$. The rod is composed of two sections of lengths $l_1$ and $l_2$ and thermal conductivities $K_1$ and $K_2$ respectively. The temperature at the interface of the two sections is

For a spherical shell with inner and outer radii $r_1$ and $r_2$ respectively,the formula for the heat flow rate is = ......... where $T_1$ and $T_2$ are the temperatures of the inner and outer surfaces of the shell $(T_1 > T_2)$. The thermal conductivity of the material of the shell is $k$.