The intensity of a point source emitting sound at distance 3 from the source is 0.707 `W//m^2?` What is the average power output of the source ?

Answer

To find the average power output of a point source emitting sound, we can use the relationship between intensity, power, and area. Here’s the step-by-step solution:

### Step 1: Understand the relationship between intensity and power
The intensity (I) of a sound wave is defined as the power (P) per unit area (A). Mathematically, this can be expressed as:
\[ I = \frac{P}{A} \]

### Step 2: Rearrange the formula to find power
From the intensity formula, we can rearrange it to find the average power:
\[ P = I \times A \]

### Step 3: Determine the area of the spherical wave front
For a point source emitting sound, the sound spreads out in a spherical wave front. The area (A) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
where \( r \) is the radius (distance from the source). In this case, \( r = 3 \) m.

### Step 4: Calculate the area
Substituting the value of \( r \):
\[ A = 4\pi (3)^2 \]
\[ A = 4\pi \times 9 \]
\[ A = 36\pi \, \text{m}^2 \]

### Step 5: Substitute the intensity and area into the power formula
We know the intensity \( I = 0.707 \, \text{W/m}^2 \). Now we can substitute the values into the power formula:
\[ P = I \times A \]
\[ P = 0.707 \times 36\pi \]

### Step 6: Calculate the power
Now we need to calculate \( P \):
\[ P \approx 0.707 \times 36 \times 3.14 \]
\[ P \approx 0.707 \times 113.04 \]
\[ P \approx 80 \, \text{W} \]

Thus, the average power output of the source is approximately **80 W**.

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