1. Use the definition of sound level

The intensity level in decibel is

$\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)$

Given:

• $\beta = 120\,\text{dB}$
• $I_0 = 10^{-12}\,\text{W/m}^2$

So,

$120 = 10 \log_{10}\left(\frac{I}{10^{-12}}\right)$

$12 = \log_{10}\left(\frac{I}{10^{-12}}\right)$

$\frac{I}{10^{-12}} = 10^{12}$

$I = 1\,\text{W/m}^2$

2. Relate intensity to power and distance

For a point source radiating uniformly,

$I = \frac{P}{4\pi r^2}$

Given power output:

$P = 2\,\text{W}$

Substitute $I = 1\,\text{W/m}^2$:

$1 = \frac{2}{4\pi r^2}$

$4\pi r^2 = 2$

$r^2 = \frac{1}{2\pi}$

$r = \sqrt{\frac{1}{2\pi}}$

3. Calculate numerically

Using $\pi \approx 3.14$,

$r = \sqrt{\frac{1}{6.28}} = \sqrt{0.159} \approx 0.399\,\text{m}$

$r \approx 40\,\text{cm}$

4. Check options

• A: $20\,\text{cm}$ ❌
• B: $10\,\text{cm}$ ❌
• C: $40\,\text{cm}$ ✅
• D: $30\,\text{cm}$ ❌

Therefore, the correct option is C.