A point source of sound radiates energy at the rate of 3.14 watts. Find the intensity level of loudness at a distance of 15.8 m from the source in dB (approximately) ?
(take `pi=3.14`)

Answer

To find the intensity level of loudness in decibels (dB) at a distance of 15.8 m from a point source of sound radiating energy at a rate of 3.14 watts, we can follow these steps:

### Step 1: Calculate the intensity of the sound
The intensity \( I \) of a sound from a point source can be calculated using the formula:
\[ I = \frac{P}{A} \]
where \( P \) is the power of the source and \( A \) is the area over which the power is distributed. For a point source, the area \( A \) is the surface area of a sphere, given by:
\[ A = 4 \pi r^2 \]
where \( r \) is the distance from the source.

### Step 2: Substitute the values
Given:
- Power \( P = 3.14 \) watts
- Distance \( r = 15.8 \) m
Using \( \pi = 3.14 \):
\[ A = 4 \times 3.14 \times (15.8)^2 \]

### Step 3: Calculate the area
First, calculate \( (15.8)^2 \):
\[ (15.8)^2 = 249.64 \]
Now, substitute this value into the area formula:
\[ A = 4 \times 3.14 \times 249.64 \approx 4 \times 3.14 \times 249.64 \approx 3140.57 \text{ m}^2 \]

### Step 4: Calculate the intensity
Now, substitute the values of \( P \) and \( A \) into the intensity formula:
\[ I = \frac{3.14}{3140.57} \approx 0.001 \text{ W/m}^2 \]

### Step 5: Convert intensity to decibels
The intensity level in decibels (dB) can be calculated using the formula:
\[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \]
where \( I_0 = 10^{-12} \text{ W/m}^2 \) is the reference intensity. Thus:
\[ L = 10 \log_{10} \left( \frac{0.001}{10^{-12}} \right) = 10 \log_{10} (10^{9}) = 10 \times 9 = 90 \text{ dB} \]

### Step 6: Finalize the answer
The intensity level of loudness at a distance of 15.8 m from the source is approximately:
\[ \text{Intensity Level} \approx 90 \text{ dB} \]