Two sheets of thickness $d$ and $2d$ and the same area are touching each other on their faces. The temperatures $T_A$,$T_B$,and $T_C$ shown are in a geometric progression with a common ratio $r = 2$. Then,the ratio of the thermal conductivity of the thinner sheet to the thicker sheet is:

$A$ room is maintained at $20^{\circ}C$ by a heater of resistance $20 \ \Omega$ connected to $200 \ V$ mains. The temperature is uniform throughout the room and heat is transmitted through a glass window of area $1 \ m^2$ and thickness $0.2 \ cm$. What will be the temperature outside in $^{\circ}C$? Given that thermal conductivity $K = 0.2 \ J/(s \cdot m \cdot K)$ for glass and $J = 4.2 \ J/cal$.

$A$ 'thermacole' icebox is a cheap and an efficient method for storing small quantities of cooked food in summer in particular. $A$ cubical icebox of side $30 \,cm$ has a thickness of $5.0 \,cm$. If $4.0 \,kg$ of ice is put in the box,estimate the amount of ice (in $kg$) remaining after $6 \,h$. The outside temperature is $45 \,^{\circ}C$,and the coefficient of thermal conductivity of thermacole is $0.01 \,J \,s^{-1} \,m^{-1} \,K^{-1}$. Heat of fusion of water $= 335 \times 10^{3} \,J \,kg^{-1}$.

In Searle's method for finding the thermal conductivity of metals,the temperature gradient along the bar:

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T_1 = 300 \ K$ and $T_2 = 100 \ K$,as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K_1$ and $K_2$ respectively. If the temperature at the junction of the two cylinders in the steady state is $200 \ K$,then $K_1 / K_2 = \dots$

Steam at $373 K$ is passed through a tube of radius $10 cm$ and length $2 m$. The thickness of the tube is $5 mm$ and the thermal conductivity of its material is $390 W m^{-1} K^{-1}$. Calculate the heat lost per second. The outside temperature is $0^{\circ}C$.