At a distance of 5.0 m from a point sound source, the sound intensity level is 110 dB. At what distance is the intensity level 95 dB?

Options

A. 5.0 m
B. 14 m
C. 7.1 m
D. 28 m

Answer

To solve the problem of finding the distance at which the sound intensity level is 95 dB, given that at a distance of 5.0 m from a point sound source the intensity level is 110 dB, we can follow these steps:

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

### Step 2: Set up the equations for the two distances
At a distance of 5.0 m, the intensity level is 110 dB:
\[ 110 = 10 \log_{10} \left( \frac{I_1}{I_0} \right) \]
At a distance \(x\) (which we need to find), the intensity level is 95 dB:
\[ 95 = 10 \log_{10} \left( \frac{I_2}{I_0} \right) \]

### Step 3: Solve for intensities \(I_1\) and \(I_2\)
From the first equation:
\[ \log_{10} \left( \frac{I_1}{I_0} \right) = 11 \implies \frac{I_1}{I_0} = 10^{11} \implies I_1 = 10^{11} I_0 \]
From the second equation:
\[ \log_{10} \left( \frac{I_2}{I_0} \right) = 9.5 \implies \frac{I_2}{I_0} = 10^{9.5} \implies I_2 = 10^{9.5} I_0 \]

### Step 4: Relate the intensities to distances
The intensity of sound from a point source is inversely proportional to the square of the distance from the source:
\[ I_1 \propto \frac{1}{r_1^2} \quad \text{and} \quad I_2 \propto \frac{1}{r_2^2} \]
Thus, we can write:
\[ \frac{I_1}{I_2} = \frac{r_2^2}{r_1^2} \]
Substituting the values we found:
\[ \frac{10^{11} I_0}{10^{9.5} I_0} = \frac{r_2^2}{(5)^2} \]
This simplifies to:
\[ 10^{1.5} = \frac{r_2^2}{25} \]

### Step 5: Solve for \(r_2\)
Rearranging gives:
\[ r_2^2 = 25 \cdot 10^{1.5} = 25 \cdot 31.6228 \approx 790.57 \]
Taking the square root:
\[ r_2 \approx \sqrt{790.57} \approx 28.14 \, \text{m} \]

### Conclusion
The distance at which the intensity level is 95 dB is approximately **28 m**.

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