Question

If a woman needs an amplification of 5.0×1012 times the threshold intensity to
enable her to hear at all frequencies, what is her overall hearing loss in dB? Note that
smaller amplification is appropriate for more intense sounds to avoid further damage to her
hearing from levels above 90 dB.

Answer 1

The sound level can be calculated with the expression

$$\beta=10\text{log}\Big(\frac{I}{I_0}\Big).$$

Since the woman needs an amplification of $5\cdot10^{12}I_0$, where $I_0$ is the threshold density, her hearing loss is

$$\beta=10\text{log}\Big(\frac{5\cdot10^{12}I_0}{I_0}\Big)=127 \text{ dB}.$$