Halliday, Resnick & Walker — Temperature, Heat, and the First Law of Thermodynamics: Problem 32

32 Penguin huddling. To withstand the harsh weather of the Antarctic, emperor penguins huddle in groups (Fig. 18-35). Assume that a penguin is a circular cylinder with a top surface area \(a = 0.26 \text{ m}^2\) and height \(h = 90 \text{ cm}\). Let \(P_r\) be the rate at which an individual penguin radiates energy to the environment (through the top and the sides); thus \(NP_r\) is the rate at which \(N\) identical, well-separated penguins radiate. If the penguins huddle closely to form a huddled cylinder with top surface area \(Na\) and height \(h\), the cylinder radiates at the rate \(P_h\). If \(N = 1000\), (a) what is the value of the fraction \(P_h/NP_r\) and (b) by what percentage does huddling reduce the total radiation loss?