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(A) The rate of heat flow through a material is given by the formula: $\frac{dQ}{dt} = \frac{KA(\Delta T)}{x}$,where $K$ is the thermal conductivity,$

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$t_{2}: t_{1}$

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(A) The rate of heat flow through a material is given by the formula: $\frac{dQ}{dt} = \frac{KA(\Delta T)}{x}$,where $K$ is the thermal conductivity,$A$ is the surface area,$\Delta T$ is the temperature difference,and $x$ is the wall thickness.Since the containers have the same size,shape,and wall thickness,$A$ and $x$ are constant. The surroundings are identical,so $\Delta T$ is also constant.For a given mass of ice $m$,the total heat required to melt it is $Q = mL$,where $L$ is the latent heat of fusion. Thus,$Q$ is constant for both containers.The rate of heat flow is inversely proportional to the time taken to melt the ice: $\frac{dQ}{dt} \propto \frac{1}{t}$.Therefore,$K \propto \frac{1}{t}$,which implies $K_P t_1 = K_Q t_2$.Rearranging this gives the ratio of thermal conductivities: $\frac{K_P}{K_Q} = \frac{t_2}{t_1}$.

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