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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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