Two spheres of different materials one with triple the radius and one-fifth wall thickness of the other are filled with ice. If the time taken for complete melting of ice in the larger sphere is 30 minute and for smaller one is 20 minute, the ratio of thermal conductivities of the materials of larger spheres to that of smaller sphere is

Solution :
Let $${K_1}$$ and $${K_2}$$ be the thermal conductivities of larger and smaller sphere.
For bigger sphere,
$$\frac{{{K_1}4\pi {{\left( {3r} \right)}^2} \times 100}}{{\frac{t}{4}}} = \frac{{\frac{4}{3}\pi {{\left( {3r} \right)}^3}\rho L}}{{30 \times 60}}$$
For smaller sphere,
$$\eqalign{ & \frac{{{K_2} \times 4\pi {r^2} \times 100}}{t} = \frac{{\frac{4}{3}\pi {r^3}\rho L}}{{20 \times 60}} \cr & \therefore \frac{{{K_1}}}{{{K_2}}} = \frac{1}{2} \cr} $$