Problem 61 - Physics

Question

When a person wears a hearing aid, the sound intensity level increases by \(30.0 \mathrm{~dB}\). By what factor does the sound intensity increase?

Step-by-Step Solution

Answer

The sound intensity increases by a factor of 1000.

Step 1: Understanding Sound Intensity and Decibels

Sound intensity level in decibels indicates how much the sound intensity increases. An increase of 30 dB implies the sound intensity has increased by a certain factor that we want to determine.

Step 2: Formula for Decibel Increase

The relationship between decibel increase (\(dB\)) and sound intensity increase is given by the formula: \( \Delta L = 10 \times \log_{10}\left(\frac{I_2}{I_1}\right) \), where \( \Delta L \) is the change in decibels, \( I_2 \) is the final intensity, and \( I_1 \) is the initial intensity.

Step 3: Substitute and Define Variables

In this problem, \(\Delta L = 30 \text{ dB}\). We rearrange the equation to solve for the intensity factor \( \frac{I_2}{I_1} \). This requires converting dB increase to a ratio: \( 30 = 10 \times \log_{10} \left(\frac{I_2}{I_1}\right) \).

Step 4: Solve for the Intensity Factor

Solve for \( \frac{I_2}{I_1} \): \( 30/10 = \log_{10} \left(\frac{I_2}{I_1}\right) \). Simplifying gives \( 3 = \log_{10} \left(\frac{I_2}{I_1}\right) \). To find the intensity factor, we use \( 10^3 = \frac{I_2}{I_1} \).

Step 5: Calculate and Write the Conclusion

The factor \( \frac{I_2}{I_1} = 10^3 = 1000 \). The sound intensity increases by a factor of 1000.

Key Concepts

decibels

Decibels (dB) are crucial in measuring sound intensity. They help us quantify how loud or soft a sound is. The decibel scale is logarithmic, meaning a small increase in dB level results in a large change in intensity.
For example, when a sound intensity level goes up by 10 dB, the actual intensity of the sound increases tenfold.
If it goes up by 20 dB, the intensity becomes 100 times greater, and at 30 dB, as in the exercise you encountered, the intensity is 1000 times greater than the initial sound level.

• This logarithmic nature makes decibels very effective for representing the vast range of sound intensities audible to humans.
• Doubling the intensity of sound corresponds to roughly a 3 dB increase.

The formula to capture this relationship is: \[ \Delta L = 10 \times \log_{10}\left(\frac{I_2}{I_1}\right) \] which helps compute the factor by which sound intensity has increased based on its dB change.

hearing aid

A hearing aid is a small electronic device that amplifies sound for individuals with hearing loss. When someone wears a hearing aid, it increases the sound intensity received by the ear.
This amplified sound helps improve speech understanding and enables the user to hear more sounds in their environment.

• Hearing aids pick up sound through a microphone, process it through an amplifier, and then deliver it to the ear via a speaker.
• For example, if a hearing aid increases the sound level by 30 dB, it makes the sounds 1000 times more intense than the natural hearing level without the aid.
• If a conversation initially has a sound intensity of 50 dB, a hearing aid taking it to 80 dB makes speech significantly clearer and more recognizable.

This functionality is achieved by converting environmental sounds into digital signals, which are then adjusted in frequency and intensity to match the user's hearing loss profile.

logarithms

Logarithms are mathematical tools that transform multiplicative relationships into additive ones, making them ideal for dealing with exponential growth.
They are especially useful in sound intensity, as sound increases exponentially with each step on the dB scale.

• The logarithm base 10, or \( \log_{10} \), is the standard in sound measurements, due to the human ear's perceptual response to sound intensity.
• Using a formula involving logarithms, the change in sound intensity can be easy to calculate.
• This process entails solving for the unknown intensity factor to understand how much louder a sound has become due to changes in decibel level.

For instance, if the sound intensity increases by 30 dB, you solve the equation \[ 3 = \log_{10} \left(\frac{I_2}{I_1}\right) \] using logarithmic properties to find that the intensity has increased by a factor of 1000.