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(C) The sound level $\beta$ in decibels $(dB)$ is given by the formula: $\beta = 10 \log_{10} \left( \frac{I}{I_{ref}} \right)$.Given the threshold of

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$10^{-12}$

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(C) The sound level $\beta$ in decibels $(dB)$ is given by the formula: $\beta = 10 \log_{10} \left( \frac{I}{I_{ref}} \right)$.Given the threshold of hearing is $0 \ dB$,we have $0 = 10 \log_{10} \left( \frac{I_0}{I_{ref}} \right)$,which implies $I_0 = I_{ref}$.Given the threshold of pain is $120 \ dB$,we have $120 = 10 \log_{10} \left( \frac{I}{I_{ref}} \right)$.Dividing by $10$,we get $12 = \log_{10} \left( \frac{I}{I_{ref}} \right)$,which implies $\frac{I}{I_{ref}} = 10^{12}$,so $I = 10^{12} I_{ref}$.Since $I_0 = I_{ref}$,we have $I = 10^{12} I_0$.Therefore,the ratio $\frac{I_0}{I} = \frac{I_0}{10^{12} I_0} = 10^{-12}$.

If the threshold of hearing is assumed to be the reference $(0 \ dB)$,then the threshold of pain is taken to be $120 \ dB$. Let the corresponding sound intensities be $I_0$ and $I$ respectively. Then $\frac{I_0}{I}$ is:

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