91Ó°ÊÓ

The area of the ear trumpet is\(900\;{\rm{c}}{{\rm{m}}^{\rm{2}}}\).

The area of the ear is\(0.500\;{\rm{c}}{{\rm{m}}^{\rm{2}}}\).

The efficiency of ear trumpet is\(5\% \).

The increase in intensity increases the loudness of a sound. The loudness is unsustainable if it is greater than the threshold level.

(a)

The intensity of the unaided hearing is,

\({d_1} = 10\log \frac{{{I_0}}}{{{{10}^{12}}}}\)

The change in intensity of sound is,

\({Q_1} = {I_0} \times 900\)

\(\begin{array}{c}{I_{ear}} = efficiency \times \frac{{{Q_1}}}{{0.500}}\\{I_{ear}} = 5.00 \times {I_0} \times \frac{{900}}{{0.500}}\\{I_{ear}} = 90{I_0}\end{array}\)

The intensity of the sound is,

\(\begin{array}{c}d = 10\log \frac{{90{I_0}}}{{{{10}^{ - 12}}}}\\d = 10\log \frac{{{I_0}}}{{{{10}^{ - 12}}}} + 10\log \frac{{90}}{{{{10}^{ - 12}}}}\\d = {d_1} + 139.5\;{\rm{dB}}\end{array}\)

Hence, the intensity increases by\(139.5\;{\rm{dB}}\)

(b)

The maximum limit for the intensity of sound is about\(139.5\;{\rm{dB}}\). The increase in the intensity level may damage the ear. The radius of the trumpet is close to \(16\;{\rm{cm}}\) which is not practical for holding the sound to the ear. This situation is not sustainable for a short period.