The normal body-temperature of a person is `97^(@)F` . Calculate the rate at which heat is flowing out of his body through the clothes assuming the following values. Room temperature `=47^(@)F` , surface of the body under clothes `=1.6m^(2)` , conductivity of the cloth `=0.04Js^(-1)m^(-1)C^(-1)` , thickness of the cloth `=0.5cm`.

Explanation

To solve the problem of calculating the rate at which heat is flowing out of a person's body through clothes, we will follow these steps:

### Step 1: Convert Temperatures from Fahrenheit to Celsius
We need to convert the body temperature and room temperature from Fahrenheit to Celsius using the formula:
\[ C = \frac{5}{9}(F - 32) \]

- **Body Temperature**:
\[ C_{body} = \frac{5}{9}(97 - 32) = \frac{5}{9}(65) \approx 36.10^\circ C \]

- **Room Temperature**:
\[ C_{room} = \frac{5}{9}(47 - 32) = \frac{5}{9}(15) \approx 8.33^\circ C \]

### Step 2: Identify Given Values
We have the following values:
- Surface area of the body under clothes, \( A = 1.6 \, m^2 \)
- Thermal conductivity of the cloth, \( k = 0.04 \, \frac{J}{s \cdot m \cdot ^\circ C} \)
- Thickness of the cloth, \( d = 0.5 \, cm = 0.005 \, m \)
- Temperature difference, \( \Delta T = C_{body} - C_{room} = 36.10 - 8.33 = 27.77^\circ C \)

### Step 3: Apply the Formula for Heat Transfer
The rate of heat transfer (\( \frac{\Delta Q}{\Delta t} \)) through conduction is given by Fourier's law:
\[ \frac{\Delta Q}{\Delta t} = \frac{k \cdot A \cdot \Delta T}{d} \]

### Step 4: Substitute the Values into the Formula
Now we can substitute the known values into the formula:
\[ \frac{\Delta Q}{\Delta t} = \frac{0.04 \, \frac{J}{s \cdot m \cdot ^\circ C} \cdot 1.6 \, m^2 \cdot 27.77 \, ^\circ C}{0.005 \, m} \]

### Step 5: Calculate the Rate of Heat Flow
Calculating the above expression:
\[ \frac{\Delta Q}{\Delta t} = \frac{0.04 \cdot 1.6 \cdot 27.77}{0.005} \]
\[ = \frac{0.044352}{0.005} = 8.8704 \, J/s \approx 356 \, W \]

### Final Answer
The rate at which heat is flowing out of the body through the clothes is approximately **356 Watts**.

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