A point source is emitting sound in all directions. The ratio of distance of two points from the point source where the difference in loudness levels is 3 dB is: `(log_(10)2=0.3)`.

Options

A. `(1)/(2)`
B. `(1)/(sqrt(2))`
C. `(1)/(4)`
D. `(2)/(3)`

Explanation

To solve the problem, we need to determine the ratio of distances \( r_1 \) and \( r_2 \) from a point source of sound where the difference in loudness levels is 3 dB.

### Step-by-Step Solution:

1. **Understanding Loudness Level**: The loudness level \( \beta \) in decibels (dB) is given by the formula:
\[ \beta = 10 \log_{10} \left( \frac{I}{I_0} \right) \]
where \( I \) is the intensity of the sound and \( I_0 \) is the reference intensity.

2. **Intensity Relation**: For a point source, the intensity \( I \) at a distance \( r \) from the source is inversely proportional to the square of the distance:
\[ I = \frac{K}{r^2} \]
where \( K \) is a constant.

3. **Substituting Intensity in Loudness Formula**: We can substitute the expression for intensity into the loudness level equation:
\[ \beta = 10 \log_{10} \left( \frac{K/r^2}{I_0} \right) = 10 \log_{10} \left( \frac{K}{I_0}} \right) - 20 \log_{10}(r) \]
Let \( \beta_1 \) be the loudness level at distance \( r_1 \) and \( \beta_2 \) at distance \( r_2 \):
\[ \beta_1 = 10 \log_{10} \left( \frac{K}{I_0} \right) - 20 \log_{10}(r_1) \]
\[ \beta_2 = 10 \log_{10} \left( \frac{K}{I_0} \right) - 20 \log_{10}(r_2) \]

4. **Finding the Difference in Loudness Levels**: The difference in loudness levels is given by:
\[ \beta_2 - \beta_1 = -20 \log_{10}(r_2) + 20 \log_{10}(r_1) = 20 \log_{10} \left( \frac{r_1}{r_2} \right) \]
We know that this difference is 3 dB:
\[ 20 \log_{10} \left( \frac{r_1}{r_2} \right) = 3 \]

5. **Solving for the Ratio**: Dividing both sides by 20 gives:
\[ \log_{10} \left( \frac{r_1}{r_2} \right) = \frac{3}{20} = 0.15 \]
Now, we can exponentiate both sides to find the ratio:
\[ \frac{r_1}{r_2} = 10^{0.15} \]

6. **Calculating the Value**: We can express \( 10^{0.15} \) using the approximation \( \log_{10} 2 \approx 0.3 \):
\[ 10^{0.15} = 10^{\frac{3}{20}} = 10^{\log_{10} \sqrt{2}} = \sqrt{2} \]
Therefore, we can express the ratio as:
\[ \frac{r_1}{r_2} = \frac{1}{\sqrt{2}} \]

### Final Answer:
The ratio of distances \( r_1 \) and \( r_2 \) where the difference in loudness levels is 3 dB is:
\[ \frac{r_1}{r_2} = \frac{1}{\sqrt{2}} \]