As a mechanical engineer, you are given two uniform metal bars $A$ and $B$, made from different metals, to determine their thermal conductivities. You measure both bars to have length $40.0 \mathrm{~cm}$ and uniform cross-sectional area $2.50 \mathrm{~cm}^{2}$. You place one end of bar $A$ in thermal contact with a very large vat of boiling water at $100.0^{\circ} \mathrm{C}$ and the other end in thermal contact with an ice-water mixture at $0.0^{\circ} \mathrm{C}$. To prevent heat loss along the bar's sides, you wrap insulation around the bar. You weigh the amount of ice initially as $300 \mathrm{~g}$. After $45.0 \mathrm{~min}$, you weigh the ice again; $191 \mathrm{~g}$ of ice remains. The ice-water mixture is in an insulated container, so the only heat entering or leaving it is the heat conducted by the metal bar.
You are confident that your data will allow you to calculate the thermal conductivity $k_{A}$ of bar $A .$ But this measurement was tedious-you don't want to repeat it for bar $B$. Instead, you glue the bars together end to end, with adhesive that has very large thermal conductivity, to make a composite bar $80.0 \mathrm{~m}$ long. You place the free end of $A$ in thermal contact with the boiling water and the free end of $B$ in thermal contact with the ice-water mixture. The composite bar is thermally insulated. Hours later, you notice that ice remains in the ice-water mixture. Measuring the temperature at the junction of the two bars, you find that it is $62.4^{\circ} \mathrm{C}$. After 10 min you repeat that measurement and get the same temperature, with ice remaining in the ice-water mixture. From your data, calculate the thermal conductivities of bar $A$ and of bar $B$.