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Chapter 16: Problem 65

A middle-aged man typically has poorer hearing than a middle-aged woman. In one case a woman can just begin to hear a musical tone, while a man can just begin to hear the tone only when its intensity level is increased by 7.8 dB relative to the just-audible intensity level for the woman. What is the ratio of the sound intensity just detected by the man to the sound intensity just detected by the woman?

Short Answer

Expert verified
The ratio of the sound intensity just detected by the man to the woman is approximately 6.03.

Step by step solution

01

Understand the Concept of Decibels

The decibel (dB) scale is a logarithmic scale used to measure sound intensity. It is defined by the formula \( L = 10 \log_{10}(I/I_0) \), where \( L \) is the sound intensity level in decibels, \( I \) is the intensity, and \( I_0 \) is the reference intensity (usually the threshold of hearing for humans).
02

Set up the Relationship Between Intensities

To find the ratio of the sound intensities, let's denote \( I_w \) as the intensity just detected by the woman, and \( I_m \) as the intensity just detected by the man. We know that \( I_m \) has to be 7.8 dB higher than \( I_w \). Hence, the relationship can be set as: \( 10 \log_{10}(I_m/I_w) = 7.8 \).
03

Solve for the Intensity Ratio

To solve for the ratio \( I_m/I_w \), we need to isolate the ratio in the equation from Step 2: \( 10 \log_{10}(I_m/I_w) = 7.8 \). Divide both sides by 10: \( \log_{10}(I_m/I_w) = 0.78 \).
04

Convert the Logarithmic Equation to a Ratio

To find \( I_m/I_w \), convert the logarithmic equation to exponential form: \( I_m/I_w = 10^{0.78} \).
05

Calculate the Ratio

Use a calculator to compute \( 10^{0.78} \), which gives approximately 6.03. So, \( I_m/I_w \approx 6.03 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decibel Scale
The decibel scale is a convenient way to express sound intensity levels. This scale is logarithmic, meaning it represents values in powers of ten.
Sound is measured in decibels (dB) using the formula: \[ L = 10 \log_{10}(I/I_0) \] where:
  • \( L \) is the sound intensity level in decibels,
  • \( I \) is the intensity of the sound,
  • \( I_0 \) is a reference intensity, typically the quietest sound that the average human ear can hear, known as the hearing threshold.
The decibel scale allows us to manage the vast range of sound intensities we encounter, from a quiet whisper to the roar of a jet engine, in a more comprehensible format.
Logarithmic Scale
The logarithmic scale, utilized in the decibel system, transforms multiplicative effects into additive effects.
This means that when sound intensity increases by a certain factor, the decibel level increases by a specific number of decibels.
For example, if the sound intensity increases tenfold, the decibel level increases by 10 dB. This is because the base of the logarithm in the formula for decibels is 10.
This approach helps to simplify calculations and comparisons when handling quantities that can vary drastically.
  • It compresses large ranges into smaller, more manageable numbers.
  • It helps us better understand differences in intensity levels, which are not easily perceived in their raw forms.
Sound Intensity
Sound intensity refers to the power carried by sound waves per unit area. It is usually measured in watts per square meter (W/m^2).
Sound intensity can vary greatly, from nearly imperceptible sounds to extremely loud noises that can harm our hearing.
The human ear responds to sound intensity logarithmically, which is why the decibel scale is so useful.
In our exercise, we compared sound intensities by calculating their ratio using decibels.
  • Sound intensity is directly related to the perceived loudness of the sound.
  • It plays a crucial role in the sensation of sound and in applications where sound needs to be quantified.
  • Finding the ratio of two sound intensities helps us understand how much more powerful one sound is compared to another.
Hearing Threshold
The hearing threshold, commonly denoted as \( I_0 \), represents the faintest sound the average human ear can detect.
It is often considered to be around \( 10^{-12} \) watts per square meter (W/m^2).
This threshold is important as it acts as a reference point for measuring other sound intensities using the decibel scale.
In our exercise, it sets the baseline for comparing the intensities heard by the man and the woman.
  • A sound at the hearing threshold has a decibel level of 0 dB, indicating the baseline.
  • Other sound intensities are measured relative to this baseline to understand how audible they are relative to the faintest sound we can hear.
  • This helps individuals and professionals in assessing hearing capability and in designing sound-producing or sound-absorbing materials.

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Most popular questions from this chapter

The average sound intensity inside a busy neighborhood restaurant is \(3.2 \times 10^{-5} \mathrm{W} / \mathrm{m}^{2}\) . How much energy goes into each ear (area \(=\) \(2.1 \times 10^{-3} \mathrm{m}^{2} )\) during a one-hour meal?

A water-skier is moving at a speed of 12.0 m/s. When she skis in the same direction as a traveling wave, she springs upward every 0.600 s because of the wave crests. When she skis in the direction opposite to the direction in which the wave moves, she springs upward every 0.500 s in response to the crests. The speed of the skier is greater than the speed of the wave. Determine (a) the speed and (b) the wavelength of the wave.

Hearing damage may occur when a person is exposed to a sound intensity level of 90.0 dB (relative to the threshold of hearing) for a period of 9.0 hours. One particular eardrum has an area of \(2.0 \times 10^{-4} \mathrm{m}^{2} .\) How much sound energy is incident on this eardrum during this time?

When an earthquake occurs, two types of sound waves are gen- erated and travel through the earth. The primary, or P, wave has a speed of about 8.0 km/s and the secondary, or S, wave has a speed of about 4.5 km/s. A seismograph, located some distance away, records the arrival of the P wave and then, 78 s later, records the arrival of the S wave. Assuming that the waves travel in a straight line, how far is the seismograph from the earthquake?

A typical adult ear has a surface area of \(2.1 \times 10^{-3} \mathrm{m}^{2}\) The sound intensity during a normal conversation is about 3.2 \(\times 10^{-6} \mathrm{W} / \mathrm{m}^{2}\) at the listener's ear. Assume that the sound strikes the surface of the ear perpendicularly. How much power is intercepted by the ear?
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