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 $0^{\circ}\,C$ of 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 conducting of insulation wall is $0.05\,Wm ^{-10} C ^{-1}$ )

Value of temperature gradient is $80\,^oC/m$ on a rod of $0.5\,m$ length. Temperature of hot end is $30\,^oC$, then what is the temperature of cold end ?

Two rods $A$ and $B$ of same cross-sectional are $A$ and length $l$ connected in series between a source $(T_1 = 100^o C)$ and a sink $(T_2 = 0^o C)$ as shown in figure. The rod is laterally insulated  If $G_A$ and $G_B$ are the temperature gradients across the rod $A$ and $B$, then 

Two metal cubes $A$ and $B$ of same size are arranged as shown in the figure. The extreme ends of the combination are maintained at the indicated temperatures. The arrangement is thermally insulated. The coefficients of thermal conductivity of $A$ and $B$ are $300\;W/m{\;^o}C$ and $200\;W/m{\;^o}C$, respectively. After steady state is reached, the temperature of the interface will be…… $^oC$

A metallic rod of cross-sectional area $9.0\,\,cm^2$ and length $0.54 \,\,m$, with the surface insulated to prevent heat loss, has one end immersed in boiling water and the other in ice-water mixture. The heat conducted through the rod melts the ice at the rate of $1 \,\,gm$ for every $33 \,\,sec$. The thermal conductivity of the rod is ……. $ Wm^{-1} K^{-1}$