The intensity of sound from a point source is `1.0 xx 10^-8 Wm^-2`, at a distance of 5.0 m from the source. What will be the intensity at a distance of 25 m from the source ?

Explanation

To solve the problem, we will use the relationship between intensity and distance for a point source of sound. The intensity \( I \) of sound is inversely proportional to the square of the distance \( r \) from the source. This can be mathematically expressed as:

\[ I \propto \frac{1}{r^2} \]

### Step 1: Identify the known values
- Initial intensity \( I_1 = 1.0 \times 10^{-8} \, \text{W/m}^2 \)
- Initial distance \( r_1 = 5.0 \, \text{m} \)
- Final distance \( r_2 = 25.0 \, \text{m} \)

### Step 2: Set up the relationship between intensities and distances
Using the inverse square law, we can write:

\[ \frac{I_1}{I_2} = \frac{r_2^2}{r_1^2} \]

### Step 3: Substitute the known values into the equation
Substituting the known values into the equation gives:

\[ \frac{1.0 \times 10^{-8}}{I_2} = \frac{(25.0)^2}{(5.0)^2} \]

### Step 4: Calculate the ratio of the distances
Calculating the squares of the distances:

\[ \frac{(25.0)^2}{(5.0)^2} = \frac{625}{25} = 25 \]

### Step 5: Rearrange the equation to solve for \( I_2 \)
Now we can rearrange the equation to solve for \( I_2 \):

\[ I_2 = \frac{1.0 \times 10^{-8}}{25} \]

### Step 6: Perform the calculation
Calculating \( I_2 \):

\[ I_2 = 4.0 \times 10^{-10} \, \text{W/m}^2 \]

### Final Answer
The intensity at a distance of 25 m from the source is:

\[ I_2 = 4.0 \times 10^{-10} \, \text{W/m}^2 \]

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