Animals in cold climates often depend on two layers of insulation: a layer of body fat (of thermal conductivity 0.20 W/m ‧ K) surrounded by a layer of air trapped inside fur or down. We can model a black bear (Ursus americanus) as a sphere 1.5 m in diameter having a layer of fat 4.0 cm thick. (Actually, the thickness varies with the season, but we are interested in hibernation, when the fat layer is thickest.) In studies of bear hibernation, it was found that the outer surface layer of the fur is at 2.7°C during hibernation, while the inner surface of the fat layer is at 31.0°C. 

(a) What is the temperature at the fat–inner fur boundary?

(b) How thick should the air layer (contained within the fur) be so that the bear loses heat at a rate of 50.0 W?