Question

If a woman needs an amplification of $5\times 10^{12}\mathrm{W}\cdot {\mathrm{m}}^{-2}$ 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$90\mathrm{d}\mathrm{B}$.

Short answer

Women’s overall hearing loss will be $122dB$

Step-by-step solution

Formula for sound intensity level in $\mathrm{d}\mathrm{B}$ (decibels):

The sound intensity level (SIL) or sound intensity level is the level (logarithmic quantity) of sound intensity relative to a reference value.

The sound intensity formula is given by,

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

Here, $\beta$ is the sound intensity level in $dB$, $I$ is the intensity (in $W\cdot m^{-2}$) of ultrasound wave and $I_0={10}^{-12}W\cdot m^{-2}$ is the threshold intensity of hearing.

Calculation for hearing loss:

Consider the given data as below.

Women need amplification of $5\times {10}^{12}W\cdot m^{-2}$ times the threshold hearing $\left(I_0\right)$.

Write the following relation of the intensity and threshold intensity as given below.

$I=5\times {10}^{12}{I}_0$

Here, the threshold intensity of hearing is,

$I_0={10}^{-12}W\cdot m^{-2}$

Calculate hearing loss $\beta$ by using following formula.

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

By replacing values of $I$ and $I_0$ you will get the following.

$\begin{array}{rl}\beta & =10{\log }_{10}\left(\frac{5\times {10}^{12}{I}_0}{I_0}\right)\\ & =10{\log }_{10}\left(5\times {10}^{12}\right)\\ & =10\times 12.2\\ & =122dB\end{array}$

Hence, Women’s overall hearing loss will be $122dB$.