To determine the intensity [tex]\( I \)[/tex] of a sound given its level [tex]\( \beta \)[/tex] in decibels (dB), we can use the decibel formula for intensity:
[tex]\[ \beta = 10 \log \left(\frac{I}{I_0}\right) \][/tex]
where [tex]\( I_0 \)[/tex] is the reference intensity, often taken to be [tex]\( 1.0 \times 10^{-12} \)[/tex] watts/m².
Given:
[tex]\[ \beta = 120 \text{ dB} \][/tex]
[tex]\[ I_0 = 1.0 \times 10^{-12} \text{ watts/m}^2 \][/tex]
We need to solve for [tex]\( I \)[/tex]. Start by isolating [tex]\( \frac{I}{I_0} \)[/tex] in the equation:
[tex]\[ 120 = 10 \log \left(\frac{I}{I_0}\right) \][/tex]
Divide both sides by 10:
[tex]\[ 12 = \log \left(\frac{I}{I_0}\right) \][/tex]
To solve for [tex]\( \frac{I}{I_0} \)[/tex], remember that if:
[tex]\[ \log x = y \][/tex]
then:
[tex]\[ x = 10^y \][/tex]
So,
[tex]\[ \frac{I}{I_0} = 10^{12} \][/tex]
Now, multiply both sides by [tex]\( I_0 \)[/tex]:
[tex]\[ I = I_0 \times 10^{12} \][/tex]
Substitute [tex]\( I_0 \)[/tex] with [tex]\( 1.0 \times 10^{-12} \)[/tex]:
[tex]\[ I = (1.0 \times 10^{-12}) \times 10^{12} \][/tex]
[tex]\[ I = 1.0 \times 10^{0} \][/tex]
[tex]\[ I = 1.0 \text{ watts/m}^2 \][/tex]
Therefore, the correct answer is:
B. [tex]\( 1.0 \times 10^0 \text{ watts/m}^2 \)[/tex]
