A person speacking normally produces a sound intensity of 40dB AT A IDISTANCE OF 1M. If the threshold intensity for reasonable audiblity 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 step by step, we will use the relationship between sound intensity levels in decibels (dB) and the corresponding intensities.

### Step 1: Understand the relationship between intensity and decibels
The sound intensity level in decibels (dB) is given by the formula:
\[ \beta = 10 \log \left( \frac{I}{I_0} \right) \]
where:
- \(\beta\) is the sound intensity level in dB,
- \(I\) is the sound intensity,
- \(I_0\) is the reference intensity, typically \(10^{-12} \, \text{W/m}^2\).

### Step 2: Calculate the intensity at 40 dB
Given that a person speaking normally produces a sound intensity of 40 dB at a distance of 1 meter, we can set up the equation:
\[ 40 = 10 \log \left( \frac{I_1}{I_0} \right) \]
Dividing both sides by 10:
\[ 4 = \log \left( \frac{I_1}{I_0} \right) \]
Exponentiating both sides gives:
\[ \frac{I_1}{I_0} = 10^4 \]
This implies:
\[ I_1 = 10^4 I_0 \]

### Step 3: Calculate the intensity at 20 dB
Next, we calculate the intensity at the threshold of reasonable audibility, which is 20 dB:
\[ 20 = 10 \log \left( \frac{I_2}{I_0} \right) \]
Dividing both sides by 10:
\[ 2 = \log \left( \frac{I_2}{I_0} \right) \]
Exponentiating both sides gives:
\[ \frac{I_2}{I_0} = 10^2 \]
This implies:
\[ I_2 = 10^2 I_0 \]

### Step 4: Find the ratio of intensities
Now we can find the ratio of the two intensities:
\[ \frac{I_1}{I_2} = \frac{10^4 I_0}{10^2 I_0} = \frac{10^4}{10^2} = 10^2 = 100 \]

### Step 5: Relate intensity to distance
The intensity of sound is inversely proportional to the square of the distance from the source:
\[ \frac{I_1}{I_2} = \frac{R_2^2}{R_1^2} \]
Where \(R_1 = 1 \, \text{m}\) (the distance at which the intensity is 40 dB) and \(R_2\) is the distance at which the intensity is 20 dB. Substituting the values we have:
\[ 100 = \frac{R_2^2}{1^2} \]
This simplifies to:
\[ R_2^2 = 100 \]
Taking the square root gives:
\[ R_2 = 10 \, \text{m} \]

### Step 6: Find the value of \(x\)
According to the problem, the maximum distance at which a person can be heard clearly is \(2x\):
\[ R_2 = 2x \]
Substituting \(R_2 = 10 \, \text{m}\):
\[ 10 = 2x \]
Dividing both sides by 2 gives:
\[ x = 5 \]

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