A chinese fire cracker when busted creates 100 dB sound at a distance of 10 cm. this level of sound is dangerous and can create deafness. To reduce the sound to a safer level, we need to stand far from it. At what minimum destance from the fire cracker a person should stand so that sound level heard by him is 40 dB? (Assume that sound energy spread equally in all directions)

Options

A. 100 m
B. 6.63 cm
C. 10 m
D. 1 m

Explanation

To solve the problem, we need to determine the minimum distance a person should stand from a Chinese firecracker to ensure that the sound level is reduced from 100 dB to 40 dB. We will use the relationship between sound intensity and sound level in decibels.

### Step-by-Step Solution:

1. **Understanding Decibel Level**:
The sound level in decibels (dB) is given by the formula:
\[ L = 10 \log_{10} \left(\frac{I}{I_0}\right) \]
where \( L \) is the sound level in dB, \( I \) is the intensity of the sound, and \( I_0 \) is the reference intensity, typically \( 10^{-12} \, \text{W/m}^2 \).

2. **Calculate Intensity at 10 cm**:
Given that the sound level is 100 dB at a distance of 10 cm (0.1 m), we can rearrange the formula to find the intensity \( I_1 \):
\[ 100 = 10 \log_{10} \left(\frac{I_1}{10^{-12}}\right) \]
Dividing both sides by 10:
\[ 10 = \log_{10} \left(\frac{I_1}{10^{-12}}\right) \]
Converting from logarithmic form:
\[ \frac{I_1}{10^{-12}} = 10^{10} \]
Thus,
\[ I_1 = 10^{10} \times 10^{-12} = 10^{-2} \, \text{W/m}^2 \]

3. **Calculate Intensity for 40 dB**:
Now, we need to find the intensity \( I_2 \) corresponding to a sound level of 40 dB:
\[ 40 = 10 \log_{10} \left(\frac{I_2}{10^{-12}}\right) \]
Dividing both sides by 10:
\[ 4 = \log_{10} \left(\frac{I_2}{10^{-12}}\right) \]
Converting from logarithmic form:
\[ \frac{I_2}{10^{-12}} = 10^{4} \]
Thus,
\[ I_2 = 10^{4} \times 10^{-12} = 10^{-8} \, \text{W/m}^2 \]

4. **Using the Inverse Square Law**:
Since sound energy spreads equally in all directions, we can use the inverse square law:
\[ \frac{I_1}{I_2} = \left(\frac{r_2}{r_1}\right)^2 \]
Where \( r_1 = 0.1 \, \text{m} \) (10 cm) and \( r_2 \) is the distance we need to find. Plugging in the intensities:
\[ \frac{10^{-2}}{10^{-8}} = \left(\frac{r_2}{0.1}\right)^2 \]
Simplifying the left side:
\[ 10^{6} = \left(\frac{r_2}{0.1}\right)^2 \]

5. **Solving for \( r_2 \)**:
Taking the square root of both sides:
\[ 10^{3} = \frac{r_2}{0.1} \]
Thus,
\[ r_2 = 0.1 \times 10^{3} = 100 \, \text{m} \]

### Conclusion:
The minimum distance a person should stand from the firecracker to ensure the sound level is reduced to 40 dB is **100 meters**.