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Chapter 14: Problem 63

A person of surface area \(1.80 \mathrm{m}^{2}\) is lying out in the sunlight to get a tan. If the intensity of the incident sunlight is $7.00 \times 10^{2} \mathrm{W} / \mathrm{m}^{2},$ at what rate must heat be lost by the person in order to maintain a constant body temperature? (Assume the effective area of skin exposed to the Sun is \(42 \%\) of the total surface area, \(57 \%\) of the incident radiation is absorbed, and that internal metabolic processes contribute another \(90 \mathrm{W}\) for an inactive person.)

Short Answer

Expert verified
To maintain a constant body temperature, the person must lose heat at a rate of approximately \(395.07 W\).

Step by step solution

01

Determining the effective skin area

First, we need to find the effective area of skin exposed to the Sun. We do this by multiplying the total surface area of the person by the given percentage of skin exposed to sunlight (42%). Effective skin area = \(1.80 m^2 \times 0.42 = 0.756 m^2\)
02

Calculating the power absorbed by the person due to sunlight

Next, we need to determine the amount of sunlight energy absorbed by the person. Given the intensity of sunlight is \(7.00 \times 10^{2} W/m^2\), we multiply it by the effective skin area and the absorption rate (57%) to find the absorbed power. Power absorbed due to sunlight = \( (0.756 m^2) \times (7.00 \times 10^{2} \dfrac{W}{m^2}) \times 0.57 = 305.0688 W\)
03

Calculating the total power absorbed by the person

Now we need to find the total power absorbed by the person. We add the power absorbed from sunlight and the power from internal metabolic processes. Total power absorbed = Power absorbed due to sunlight + Internal metabolic power = \(305.0688 W + 90 W = 395.0688 W\)
04

Finding the rate of heat loss required to maintain the constant body temperature

As the person should maintain a constant body temperature, the total power absorbed by the person must be equal to the power lost by the person. So, the required rate of heat loss is: Required rate of heat loss = Total power absorbed by the person = \(395.0688 W\) In order to maintain a constant body temperature while lying out in the sunlight, the person must lose heat at a rate of approximately \(395.07 W\).

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