The noise level in a classroom in absence of the teacher is 50 dB when 50 students are present. Assuming that on the average each student outputs same sound energy per second, what will be the noise level if the number of students is increased to 100 ?

Explanation

To solve the problem step by step, we will use the relationship between sound intensity and decibel level.

### Step 1: Understand the Decibel Formula
The sound level in decibels (dB) is given by the formula:
\[ \beta = 10 \log_{10} \left( \frac{I}{I_0} \right) \]
where:
- \(\beta\) is the sound level in decibels,
- \(I\) is the intensity of the sound,
- \(I_0\) is the reference intensity (usually taken as \(10^{-12} \, \text{W/m}^2\)).

### Step 2: Calculate the Intensity for 50 Students
Given that the noise level is 50 dB when there are 50 students, we can set up the equation:
\[ 50 = 10 \log_{10} \left( \frac{50I}{I_0} \right) \]
Dividing both sides by 10 gives:
\[ 5 = \log_{10} \left( \frac{50I}{I_0} \right) \]
Now, we can rewrite this in exponential form:
\[ \frac{50I}{I_0} = 10^5 \]
Thus, the total intensity when there are 50 students is:
\[ 50I = 10^5 I_0 \]

### Step 3: Calculate the Intensity for 100 Students
When the number of students increases to 100, the total intensity becomes:
\[ 100I \]
Now we need to find the new noise level \(\beta_{100}\):
\[ \beta_{100} = 10 \log_{10} \left( \frac{100I}{I_0} \right) \]
This can be expressed as:
\[ \beta_{100} = 10 \log_{10} \left( \frac{100I}{I_0} \right) = 10 \log_{10} \left( \frac{2 \cdot 50I}{I_0} \right) \]

### Step 4: Use Logarithmic Properties
Using the property of logarithms:
\[ \log_{10}(ab) = \log_{10}(a) + \log_{10}(b) \]
we can rewrite \(\beta_{100}\):
\[ \beta_{100} = 10 \left( \log_{10}(2) + \log_{10}\left(\frac{50I}{I_0}\right) \right) \]
Since we know \(\beta_{50} = 50\) dB, we can substitute:
\[ \beta_{100} = 10 \log_{10}(2) + 50 \]

### Step 5: Calculate \(\log_{10}(2)\)
Using the approximate value \(\log_{10}(2) \approx 0.301\):
\[ \beta_{100} = 10 \times 0.301 + 50 = 3.01 + 50 = 53.01 \, \text{dB} \]

### Step 6: Final Answer
Rounding to two decimal places, the noise level when there are 100 students is approximately:
\[ \beta_{100} \approx 53 \, \text{dB} \]

### Summary
The noise level in a classroom with 100 students is approximately 53 dB.

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