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Chapter 14: Problem 53

The thermal conductivity of the fur (including the skin) of a male Husky dog is \(0.026 \mathrm{W} /(\mathrm{m} \cdot \mathrm{K}) .\) The dog's heat output is measured to be \(51 \mathrm{W}\), its internal temperature is $38^{\circ} \mathrm{C},\( its surface area is \)1.31 \mathrm{m}^{2},$ and the thickness of the fur is \(5.0 \mathrm{cm} .\) How cold can the outside temperature be before the dog must increase its heat output?

Short Answer

Expert verified
The minimum outside temperature before the dog must increase its heat output is approximately -36.9°C.

Step by step solution

01

Convert the thickness of the fur to meters

The thickness of the fur is given in centimeters. Convert it to meters: \(d = 5.0 \mathrm{cm} = 0.05 \mathrm{m}\)
02

Identify the known variables

The known variables are: - Thermal conductivity, \(k = 0.026 \mathrm{W} /(\mathrm{m} \cdot \mathrm{K})\) - Heat output, \(Q = 51 \mathrm{W}\) - Internal temperature, \(T_{hot} = 38^{\circ}\mathrm{C}\) - Surface area, \(A = 1.31 \mathrm{m}^2\) - Thickness of the fur, \(d = 0.05 \mathrm{m}\)
03

Rearrange the heat transfer formula to solve for \(T_{cold}\)

We will rearrange the formula to solve for the temperature of the colder side: \(T_{cold} = T_{hot} - \frac{Q * d}{k * A}\)
04

Substitute the known values and solve for \(T_{cold}\)

Now, plug the known values into the formula and solve for the outside temperature: \(T_{cold} = 38 - \frac{51 * 0.05}{0.026 * 1.31} = 38 - \frac{2.55}{0.03406} \approx 38 - 74.9\) \(T_{cold} \approx -36.9^{\circ}\mathrm{C}\) So, the outside temperature can be as low as approximately \(-36.9^{\circ}\mathrm{C}\) before the Husky dog must increase its heat output.

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Most popular questions from this chapter

What is the heat capacity of a system consisting of (a) a \(0.450-\mathrm{kg}\) brass cup filled with \(0.050 \mathrm{kg}\) of water? (b) \(7.5 \mathrm{kg}\) of water in a 0.75 -kg aluminum bucket?
A cylinder contains 250 L of hydrogen gas \(\left(\mathrm{H}_{2}\right)\) at \(0.0^{\circ} \mathrm{C}\) and a pressure of 10.0 atm. How much energy is required to raise the temperature of this gas to \(25.0^{\circ} \mathrm{C} ?\)
A copper bar of thermal conductivity $401 \mathrm{W} /(\mathrm{m} \cdot \mathrm{K})\( has one end at \)104^{\circ} \mathrm{C}$ and the other end at \(24^{\circ} \mathrm{C}\). The length of the bar is \(0.10 \mathrm{m}\) and the cross-sectional area is \(1.0 \times 10^{-6} \mathrm{m}^{2} .\) (a) What is the rate of heat conduction, \(\mathscr{P}\) along the bar? (b) What is the temperature gradient in the bar? (c) If two such bars were placed in series (end to end) between the same temperature baths, what would \(\mathscr{P}\) be? (d) If two such bars were placed in parallel (side by side) with the ends in the same temperature baths, what would \(\mathscr{P}\) be? (e) In the series case, what is the temperature at the junction where the bars meet?
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