A person speaking normally produces a sound intensity of 40dB at a distance of 1m. If the threshold intensity for reasonable audibility is 20dB, the maximum distance at which a person can be heard clearly is (2x) meter . Find the value of x.

Answer

To solve the problem, we need to find the maximum distance at which a person can be heard clearly, given the sound intensity levels at different distances. Here’s a step-by-step solution:

### Step 1: Understand the Given Information
- The sound intensity level produced by a person speaking normally is 40 dB at a distance of 1 meter.
- The threshold intensity for reasonable audibility is 20 dB.
- We need to find the maximum distance (2x) at which the sound can be heard clearly.

### Step 2: Use the Formula for Sound Intensity Level
The sound intensity level in decibels (dB) is given by the formula:
\[ \beta = 10 \log_{10} \left( \frac{I}{I_0} \right) \]
where:
- \( \beta \) is the sound intensity level in dB,
- \( I \) is the sound intensity,
- \( I_0 \) is the reference intensity, which is \( 10^{-12} \, \text{W/m}^2 \).

### Step 3: Set Up the Equations for the Two Intensity Levels
1. For the first intensity level (40 dB):
\[ 40 = 10 \log_{10} \left( \frac{I_1}{I_0} \right) \quad \text{(Equation 1)} \]
2. For the second intensity level (20 dB):
\[ 20 = 10 \log_{10} \left( \frac{I_2}{I_0} \right) \quad \text{(Equation 2)} \]

### Step 4: Solve for Intensities \( I_1 \) and \( I_2 \)
From Equation 1:
\[ \log_{10} \left( \frac{I_1}{I_0} \right) = 4 \implies \frac{I_1}{I_0} = 10^4 \implies I_1 = 10^4 I_0 \]

From Equation 2:
\[ \log_{10} \left( \frac{I_2}{I_0} \right) = 2 \implies \frac{I_2}{I_0} = 10^2 \implies I_2 = 10^2 I_0 \]

### Step 5: Relate the Intensities to Distances
The intensity of sound is inversely proportional to the square of the distance:
\[ I \propto \frac{1}{R^2} \]
Thus, we can write:
\[ \frac{I_1}{I_2} = \frac{R_2^2}{R_1^2} \]
Substituting the values of \( I_1 \) and \( I_2 \):
\[ \frac{10^4 I_0}{10^2 I_0} = \frac{R_2^2}{R_1^2} \]
This simplifies to:
\[ 100 = \frac{R_2^2}{R_1^2} \implies R_2^2 = 100 R_1^2 \]
Taking the square root:
\[ R_2 = 10 R_1 \]

### Step 6: Substitute the Known Distance
Given \( R_1 = 1 \, \text{m} \):
\[ R_2 = 10 \times 1 = 10 \, \text{m} \]

### Step 7: Find the Value of \( x \)
Since the maximum distance \( R_2 \) is given as \( 2x \):
\[ 2x = 10 \implies x = \frac{10}{2} = 5 \, \text{m} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{5} \]