Four sources of sound each of sound level `10 dB` are sounded together in phase , the resultant intensity level will be `(log _(10) 2 = 0.3)`

Options

A. `40 dB`
B. `26 dB`
C. `22 dB`
D. `13 dB`

Explanation

To solve the problem of finding the resultant intensity level when four sources of sound, each at a sound level of 10 dB, are sounded together in phase, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Sound Levels**:
Each sound source has a sound level of 10 dB. The sound level in decibels (dB) is related to the intensity of sound.

2. **Intensity Relationship**:
The intensity level \( L \) in dB is given by the formula:
\[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \]
where \( I \) is the intensity of the sound and \( I_0 \) is the reference intensity.

3. **Calculating Intensity for One Source**:
For one source at 10 dB:
\[ 10 = 10 \log_{10} \left( \frac{I_1}{I_0} \right) \]
Dividing both sides by 10:
\[ 1 = \log_{10} \left( \frac{I_1}{I_0} \right) \]
Therefore,
\[ \frac{I_1}{I_0} = 10 \quad \Rightarrow \quad I_1 = 10 I_0 \]

4. **Calculating Total Intensity for Four Sources**:
Since the sources are in phase, the intensities add up. Thus, the total intensity \( I_{total} \) from four sources is:
\[ I_{total} = 4 I_1 = 4 \times 10 I_0 = 40 I_0 \]

5. **Finding Resultant Sound Level**:
Now we can find the resultant sound level \( L_{total} \):
\[ L_{total} = 10 \log_{10} \left( \frac{I_{total}}{I_0} \right) = 10 \log_{10} \left( \frac{40 I_0}{I_0} \right) = 10 \log_{10} (40) \]

6. **Using Logarithm Properties**:
We can express 40 as \( 10 \times 4 \):
\[ \log_{10} (40) = \log_{10} (10) + \log_{10} (4) = 1 + \log_{10} (4) \]
Using the approximation \( \log_{10} (4) = 2 \log_{10} (2) \) and given \( \log_{10} (2) = 0.3 \):
\[ \log_{10} (4) = 2 \times 0.3 = 0.6 \]
Thus,
\[ \log_{10} (40) = 1 + 0.6 = 1.6 \]

7. **Calculating Final Sound Level**:
Therefore,
\[ L_{total} = 10 \times 1.6 = 16 \text{ dB} \]
Since we need to add the original sound level of 10 dB:
\[ L_{final} = 10 + 16 = 26 \text{ dB} \]

### Final Answer:
The resultant intensity level when four sources of sound each at 10 dB are sounded together in phase is **26 dB**.