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Expert-verified Found in: Page 595 ### College Physics (Urone)

Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000 # (a) Ear trumpets were never very common, but they did aid people with hearing losses by gathering sound over a large area and concentrating it on the smaller area of the eardrum. What decibel increase does an ear trumpet produce if its sound gathering area is$$900\;{\rm{c}}{{\rm{m}}^{\rm{2}}}$$and the area of the eardrum is$$0.500\;{\rm{c}}{{\rm{m}}^{\rm{2}}}$$, but the trumpet only has an efficiency of $$5.00\%$$in transmitting the sound to the eardrum? (b) Comment on the usefulness of the decibel increase found in part (a)

(a) The sound increases by$$139.5\;{\rm{dB}}$$

(b) It is not sustainable in the longer term but it will amplify the sound.

See the step by step solution

## Given Data

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$$5\%$$.

## Loudness of Sound

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

## Calculation of the intensity of the first sound

(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$$139.5\;{\rm{dB}}$$

## Usefulness of the situation

(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. ### Want to see more solutions like these? 