A person is talking in a small room and the sound intensity level is 60dB everywhere within the room. If there ar eight people talking simultaneously in the room, what is the sound intensity level?

Options

A. 96dB
B. 69dB
C. 74dB
D. 81dB

Answer

To solve the problem of determining the sound intensity level when eight people are talking simultaneously in a room where one person talking produces a sound intensity level of 60 dB, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial Sound Intensity Level**:
- We know that the sound intensity level (L) due to one person talking is given as 60 dB.

2. **Defining the Intensity**:
- Let \( I \) be the intensity due to one person talking.
- The intensity level in decibels is given by the formula:
\[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \]
- Here, \( I_0 \) is the reference intensity, typically \( 10^{-12} \, \text{W/m}^2 \).

3. **Calculating the Intensity for Eight People**:
- If one person has an intensity \( I \), then the intensity when eight people are talking simultaneously is:
\[ I' = 8I \]

4. **Finding the New Sound Intensity Level**:
- The new sound intensity level \( L' \) can be calculated using the formula:
\[ L' = 10 \log_{10} \left( \frac{I'}{I_0} \right) \]
- Substituting \( I' = 8I \):
\[ L' = 10 \log_{10} \left( \frac{8I}{I_0} \right) \]

5. **Using Logarithmic Properties**:
- We can separate the logarithm:
\[ L' = 10 \log_{10} (8) + 10 \log_{10} \left( \frac{I}{I_0} \right) \]
- Since \( L = 10 \log_{10} \left( \frac{I}{I_0} \right) = 60 \, \text{dB} \), we can substitute:
\[ L' = 10 \log_{10} (8) + 60 \]

6. **Calculating \( \log_{10} (8) \)**:
- We know that \( 8 = 2^3 \), so:
\[ \log_{10} (8) = 3 \log_{10} (2) \]
- The approximate value of \( \log_{10} (2) \) is about 0.301:
\[ \log_{10} (8) \approx 3 \times 0.301 = 0.903 \]

7. **Substituting Back**:
- Now substituting back into the equation for \( L' \):
\[ L' = 10 \times 0.903 + 60 \]
\[ L' \approx 9.03 + 60 = 69.03 \, \text{dB} \]

8. **Final Result**:
- Rounding to the nearest whole number, the sound intensity level when eight people are talking is approximately:
\[ L' \approx 69 \, \text{dB} \]

### Conclusion:
The sound intensity level when eight people are talking simultaneously in the room is approximately **69 dB**.