A person speaking normally produces a sound intensity of `40 dB` at a distance of `1 m`. If the threshold intensity for reasonable audibility is `20 dB`, the maximum distance at which he can be heard cleary is.

Options

A. 4 m
B. 5 m
C. 10 m
D. 20 m

Explanation

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the given data
- The sound intensity level produced by the person at a distance of 1 meter is \(40 \, \text{dB}\).
- The threshold intensity for reasonable audibility is \(20 \, \text{dB}\).
- We need to find the maximum distance 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 \(I\) is the intensity of the sound, and \(I_0\) is the reference intensity, \(I_0 = 10^{-12} \, \text{W/m}^2\).

### Step 3: Calculate the intensity at 40 dB
Using the formula for \(40 \, \text{dB}\):
\[ 40 = 10 \log_{10} \left(\frac{I_1}{10^{-12}}\right) \]
Dividing both sides by 10:
\[ 4 = \log_{10} \left(\frac{I_1}{10^{-12}}\right) \]
Exponentiating both sides to remove the logarithm:
\[ \frac{I_1}{10^{-12}} = 10^4 \]
Thus,
\[ I_1 = 10^4 \times 10^{-12} = 10^{-8} \, \text{W/m}^2 \]

### Step 4: Calculate the intensity at 20 dB
Using the formula for \(20 \, \text{dB}\):
\[ 20 = 10 \log_{10} \left(\frac{I_2}{10^{-12}}\right) \]
Dividing both sides by 10:
\[ 2 = \log_{10} \left(\frac{I_2}{10^{-12}}\right) \]
Exponentiating both sides:
\[ \frac{I_2}{10^{-12}} = 10^2 \]
Thus,
\[ I_2 = 10^2 \times 10^{-12} = 10^{-10} \, \text{W/m}^2 \]

### Step 5: Relate intensities to distances
We know that intensity is inversely proportional to the square of the distance:
\[ \frac{I_2}{I_1} = \frac{R_1^2}{R_2^2} \]
Where \(R_1 = 1 \, \text{m}\) (the distance at which \(I_1\) is measured).

### Step 6: Substitute known values
Substituting the values of \(I_1\) and \(I_2\):
\[ \frac{10^{-10}}{10^{-8}} = \frac{1^2}{R_2^2} \]
This simplifies to:
\[ 10^{-2} = \frac{1}{R_2^2} \]
Taking the reciprocal:
\[ R_2^2 = 10^2 \]
Taking the square root:
\[ R_2 = 10 \, \text{m} \]

### Conclusion
The maximum distance at which the person can be heard clearly is \(10 \, \text{m}\).

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