(Solved): please clear answer Animals in cold climates often depend on two layers of insulation: a laye ...

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Animals in cold climates often depend on two layers of insulation: a layer of body fat (of thermal conductivity \( 0.200 \mathrm{~W} /(\mathrm{m} \cdot \mathrm{K}) \) ) surrounded by a layer of air trapped inside fur or down. We can model a black bear as a sphere \( 1.50 \mathrm{~m} \) in diameter having a layer of fat \( 4.00 \times 10^{-2} \mathrm{~m} \) thick, as shown in the figure. (Figure 1) In studies of bear hibernation, it was found that the outer surface layer of the fur is at \( T_{\mathrm{C}}=2.70{ }^{\circ} \mathrm{C} \) during hibernation, while the inner surface of the fat layer is at \( T_{\mathrm{H}}=31.0^{\circ} \mathrm{C} \). How thick should the air layer (contained within the fur) be so that the bear loses heat at a rate of \( 50.0 \mathrm{~W} \) ? Thermal conductivity of air is \( k_{\text {air }} \) \( =2.40 \times 10^{-2} \mathrm{~W} /(\mathrm{m} \cdot \mathrm{K}) \). How many inches thick should the bear's fur layer \( \left(L_{\text {fur }}\right) \) be to hold a layer of air \( L_{\text {air }} \) so that adequate insulation is provided? Express your answer to three significant figures. What is the thickness of the air layer (contained within the fur), \( L_{\text {air }} \) ? Express your answer in meters. View Available Hint(s)