A baby's mouth is 30 cm from her father's ear and 1.50 m from her mother's ear. What is the difference between the sound intensity levels heard by the father and by the mother?

Verified step by step guidance

1. Identify the formula for sound intensity level (in decibels, dB), which is given by $L = 10 \log\left(\frac{I}{I_0}\right)$, where $I$ is the sound intensity and $I_0$ is the reference intensity.
2. Recognize that sound intensity, $I$, decreases with the square of the distance from the source, according to the inverse square law: $I \propto \frac{1}{r^2}$, where $r$ is the distance from the source.
3. Calculate the ratio of the intensities at the father's and mother's locations by using the distances given. Since intensity is inversely proportional to the square of the distance, the ratio of intensities can be expressed as $\frac{I_{father}}{I_{mother}} = \left(\frac{r_{mother}}{r_{father}}\right)^2$.
4. Substitute the distances into the ratio formula to find $\frac{I_{father}}{I_{mother}}$. Use $r_{father} = 0.30 \, \text{m}$ and $r_{mother} = 1.50 \, \text{m}$.
5. Calculate the difference in sound intensity levels using the formula $\Delta L = 10 \log\left(\frac{I_{father}}{I_{mother}}\right)$. This will give you the difference in decibels between what the father and the mother hear.

Key Concepts

Sound Intensity Level

Sound intensity level is a measure of the power per unit area carried by a sound wave, typically expressed in decibels (dB). It quantifies how loud a sound is perceived by the human ear, with a reference level of 0 dB corresponding to the threshold of hearing. The intensity level increases logarithmically with the increase in sound intensity, meaning that a small increase in intensity can result in a significant increase in decibels.

Inverse Square Law

The inverse square law states that the intensity of sound decreases with the square of the distance from the source. This means that if you double the distance from the sound source, the intensity of the sound is reduced to one-fourth. This principle is crucial for understanding how sound levels differ at varying distances, such as between the father and mother in the given scenario.

Decibel Calculation

To calculate the difference in sound intensity levels in decibels, the formula used is L = 10 log10(I/I0), where L is the sound level in decibels, I is the intensity of the sound, and I0 is the reference intensity. When comparing two sound levels, the difference can be calculated using the distances from the sound source to each listener, applying the inverse square law to find the respective intensities before converting to decibels.