Consider two insulating sheets with thermal resistances $R_1$ and $R_2$ as shown in the figure. The temperature $\theta$ at the junction is:

Three conducting rods of the same material and cross-section are shown in the figure. The temperatures at $A$, $D$, and $C$ are maintained at $20 ^oC$, $90 ^oC$, and $0 ^oC$ respectively. If there is no heat flow in $AB$, find the ratio of the lengths of $BD$ and $BC$.

Four identical conducting rods are joined as shown in the figure. Points $A$ and $D$ are maintained at temperatures of $200^{\circ}C$ and $20^{\circ}C$ respectively. The temperature of junction $B$ will be ....... $^{\circ}C$.

$A$ composite slab is made of two different metals of the same thickness,having thermal conductivities $K$ and $2K$. The equivalent thermal conductivity of the composite slab is:

$A$ copper rod of length $75 \ cm$ and an iron rod of length $125 \ cm$ are joined together end to end. Both are of circular cross-section with diameter $2 \ cm$. The free ends of the copper and iron are maintained at $100^{\circ} C$ and $0^{\circ} C$ respectively. The surfaces of the bars are insulated thermally. The temperature of the copper-iron junction is (Thermal conductivity of copper is $386.4 \ W/m-K$ and thermal conductivity of iron is $48.46 \ W/m-K$) (in $^{\circ} C$)

Why do people prefer to give a layer of sand or foam insulation on ceilings made of concrete?