Final answer:
The rate of heat conduction through the animal's fur is approximately 32.579 watts given the differences between the animal's skin temperature and the air temperature. Over a day, this amounts to a heat loss of 2,814,861.6 joules, which translates to a dietary energy requirement of about 673 kcal to replace this heat loss.
Step-by-step explanation:
Rate of Heat Conduction and Food Intake for Heat Replacement-
To determine the rate of heat conduction through the animal's fur, we can use Fourier's law of heat conduction. This law states that the heat transfer rate (Q) through a material is proportional to the temperature difference (ΔT) across the material and the area (A) through which heat is being transferred, and inversely proportional to the thickness (L) of the material. The formula is given by: Q = kA(ΔT/L), where k is the thermal conductivity.
Given that the animal's skin temperature is 32.0°C and the air temperature is -5.00°C, the temperature difference is 32.0°C - (-5.00°C) = 37.0°C. The surface area through which heat is being transferred is 1.40 m², the thickness of the fur is 3.14 cm (or 0.0314 m), and the thermal conductivity (k) is that of air, which is approximately 0.025 W/m°C.
The rate of heat conduction is therefore: Q = 0.025 W/m°C * 1.40 m² * (37.0°C / 0.0314 m) = 32.579 W (watts).
For part (b), assuming that the rate of heat loss is constant over a 24-hour period, the total heat loss in one day would be 32.579 W * 86,400 s/day = 2,814,861.6 J/day. As 1 dietary calorie (kcal) is equal to approximately 4,184 J, the energy needed from food to replace this heat loss would be 2,814,861.6 J/day / 4,184 J/kcal ≈ 672.7 kcal/day.
Therefore, the animal would need an intake of about 673 kcal of food per day to replace the heat transferred through conduction.