In the Ingen-Hauz's experiment,the wax melts up to lengths $10 \ cm$ and $25 \ cm$ on two identical rods of different materials. The ratio of thermal conductivities of the two materials is:

$A$ 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 $^{\circ} C$)

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 $\left(L_{ice}=80 \ cal \ g^{-1}\right)$.

If $4 \, kg$ of ice is inside a closed cubical thermocol box of side length $20 \, cm$ and wall thickness $4 \, cm$, then the mass of the ice remaining after $10 \, hours$ is nearly. (Given: Outside temperature $= 50^{\circ}C$, Thermal conductivity of thermocol $K = 0.01 \, Js^{-1}m^{-1\circ}C^{-1}$, Latent heat of fusion of ice $L = 335 \times 10^3 \, Jkg^{-1}$) (in $ \, kg$)

There are two identical vessels filled with equal amounts of ice. The vessels are made of different metals. If the ice melts in the two vessels in $20$ and $35$ minutes respectively,the ratio of the coefficients of thermal conductivity of the two metals is:

Three rods of identical cross-section and lengths are made of three different materials of thermal conductivity $K_{1}, K_{2},$ and $K_{3}$,respectively. They are joined together at their ends to make a long rod. One end of the long rod is maintained at $100^{\circ} C$ and the other at $0^{\circ} C$. If the joints of the rod are at $70^{\circ} C$ and $20^{\circ} C$ in steady state and there is no loss of energy from the surface of the rod,the correct relationship between $K_{1}, K_{2}$ and $K_{3}$ is: