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/sqrt2`
C. `1/4`
D. `2/3`

Answer

To solve the problem, we need to find the ratio of the distances of two points from a point source where the difference in loudness levels is 3 dB.

### Step-by-Step Solution:

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

2. **Intensity and Distance Relationship**:
The intensity \( I \) of sound from a point source is inversely proportional to the square of the distance \( r \) from the source:
\[ I \propto \frac{1}{r^2} \]
This means we can express the intensity as:
\[ I = \frac{k}{r^2} \]
where \( k \) is a constant.

3. **Expressing Loudness in Terms of Distance**:
Substituting the expression for intensity into the loudness formula, we get:
\[ L = 10 \log_{10} \left( \frac{k}{r^2} \cdot \frac{1}{I_0} \right) = 10 \log_{10} \left( \frac{k}{I_0} \right) - 20 \log_{10} (r) \]
Let \( K' = \frac{k}{I_0} \), then:
\[ L = 10 \log_{10} K' - 20 \log_{10} r \]

4. **Loudness Levels at Points A and B**:
Let \( R_A \) and \( R_B \) be the distances from the source to points A and B, respectively. The loudness levels at these points are:
\[ L_A = 10 \log_{10} K' - 20 \log_{10} R_A \]
\[ L_B = 10 \log_{10} K' - 20 \log_{10} R_B \]

5. **Finding the Difference in Loudness Levels**:
The difference in loudness levels is given as:
\[ L_A - L_B = 3 \text{ dB} \]
Substituting the expressions for \( L_A \) and \( L_B \):
\[ (10 \log_{10} K' - 20 \log_{10} R_A) - (10 \log_{10} K' - 20 \log_{10} R_B) = 3 \]
Simplifying this gives:
\[ -20 \log_{10} R_A + 20 \log_{10} R_B = 3 \]
Dividing through by 20:
\[ \log_{10} R_B - \log_{10} R_A = \frac{3}{20} \]

6. **Using Properties of Logarithms**:
This can be rewritten using the properties of logarithms:
\[ \log_{10} \left( \frac{R_B}{R_A} \right) = \frac{3}{20} \]
Exponentiating both sides gives:
\[ \frac{R_B}{R_A} = 10^{\frac{3}{20}} \]

7. **Substituting the Given Value**:
We know from the problem statement that \( \log_{10} 2 = 0.3 \). Therefore:
\[ 10^{\frac{3}{20}} = 10^{0.15} = 10^{\frac{1}{2} \cdot 0.3} = (10^{0.3})^{\frac{1}{2}} = 2^{\frac{1}{2}} = \sqrt{2} \]
Thus, we have:
\[ \frac{R_B}{R_A} = \sqrt{2} \]

8. **Finding the Final Ratio**:
Therefore, the ratio of the distances is:
\[ \frac{R_A}{R_B} = \frac{1}{\sqrt{2}} \]

### Final Answer:
The ratio of the distances of the two points from the point source is:
\[ \frac{R_A}{R_B} = \frac{1}{\sqrt{2}} \]