Four souces of sound each of sound level 10dB are sounded together in phase, the resultant intensity level will be (110/n) dB. Find value of n `(log_(10)2=0.3)`.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have four sound sources, each with a sound level of 10 dB. When they are sounded together in phase, we need to find the resultant intensity level in the form of \( \frac{110}{n} \) dB and determine the value of \( n \).

### Step 2: Convert Decibels to Intensity
The sound level in decibels (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 (usually \( 10^{-12} \, \text{W/m}^2 \)).

For each source:
\[ 10 = 10 \log_{10} \left( \frac{I}{I_0} \right) \]
Dividing both sides by 10:
\[ 1 = \log_{10} \left( \frac{I}{I_0} \right) \]
Taking the antilogarithm:
\[ \frac{I}{I_0} = 10 \implies I = 10 I_0 \]

### Step 3: Calculate the Total Intensity
Since we have four sources, the total intensity when they are in phase is:
\[ I_{\text{total}} = 4I = 4 \times 10 I_0 = 40 I_0 \]

### Step 4: Convert Total Intensity Back to Decibels
Now, we convert the total intensity back to decibels:
\[ L_{\text{total}} = 10 \log_{10} \left( \frac{I_{\text{total}}}{I_0} \right) = 10 \log_{10} \left( \frac{40 I_0}{I_0} \right) = 10 \log_{10}(40) \]

### Step 5: Simplify \( \log_{10}(40) \)
We can express 40 as \( 2^3 \times 5 \):
\[ \log_{10}(40) = \log_{10}(2^3) + \log_{10}(5) = 3 \log_{10}(2) + \log_{10}(5) \]
Using the approximation \( \log_{10}(2) \approx 0.3 \) and \( \log_{10}(5) \approx 0.7 \) (since \( \log_{10}(10) = 1 \)):
\[ \log_{10}(40) \approx 3(0.3) + 0.7 = 0.9 + 0.7 = 1.6 \]

### Step 6: Calculate the Total Loudness
Now substituting back into the loudness equation:
\[ L_{\text{total}} = 10 \times 1.6 = 16 \, \text{dB} \]

### Step 7: Find the Resultant Intensity Level
The total intensity level is:
\[ L_{\text{total}} = 10 + 16 = 26 \, \text{dB} \]

### Step 8: Set Up the Equation
According to the problem, we have:
\[ \frac{110}{n} = 26 \]
Multiplying both sides by \( n \):
\[ 110 = 26n \]
Dividing both sides by 26:
\[ n = \frac{110}{26} \approx 4.23 \]

### Step 9: Final Calculation
Since \( n \) must be a whole number, we round it to the nearest integer, which gives \( n = 5 \).

### Final Answer
The value of \( n \) is \( 5 \).

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