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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 \(7.8 \mathrm{dB}\) 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 sound intensity just detected by the man to the woman is approximately 6.03.

Step by step solution

01

Understanding dB Increase

The difference in intensity levels due to the increase is given as 7.8 dB. Since decibels (dB) is a logarithmic unit used to describe a ratio, this means the man's hearing threshold is 7.8 dB higher than the woman's.
02

Decibel Formula Preliminary Setup

The formula to convert intensity differences into a decibel measurement is: \[ ext{dB} = 10 \log_{10}\left(\frac{I_m}{I_w}\right)\]where \(I_m\) is the intensity detectable by the man, and \(I_w\) is the intensity detectable by the woman. Given that dB = 7.8, we can set up the equation accordingly.
03

Substitute Known Values into Formula

We know that the difference is 7.8 dB, so substitute 7.8 into the formula:\[7.8 = 10 \log_{10}\left(\frac{I_m}{I_w}\right)\]
04

Isolate Logarithmic Term

Divide both sides by 10 to isolate the logarithmic term:\[\log_{10}\left(\frac{I_m}{I_w}\right) = 0.78\]
05

Solve for Intensity Ratio

To eliminate the logarithm, convert the equation to the exponential form:\[\frac{I_m}{I_w} = 10^{0.78}\]
06

Calculate the Intensity Ratio

Calculate \(10^{0.78}\) to find the ratio:\[\frac{I_m}{I_w} \approx 6.03\]

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Key Concepts

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

Decibel
The decibel (dB) is a unit of measurement often used to express sound intensity levels. It might come across as a bit complex because it is a logarithmic unit. This simply means that it measures ratios on a scale that compresses large numerical differences to be manageable. Decibels are especially useful in acoustics, as they help in comparing the intensity of sounds, such as in the exercise with the man and the woman's hearing abilities.

  • Decibels compare the intensity of two sounds.
  • They are a relative unit, indicating how much more or less intense one sound is compared to another.
  • A small increase in decibels can represent a large increase in actual sound intensity.
Logarithmic Scale
A logarithmic scale is a nonlinear scale that is used when there is a large range of quantities to show. In acoustics and many scientific fields, a logarithmic scale helps handle values that span several orders of magnitude. This makes decibels a suitable choice for measuring sound levels.

  • The logarithmic scale compresses the extensive range of sound intensities that humans can hear into a manageable range.
  • This is why an increase of 10 dB means the actual intensity increases by a factor of 10.
  • In the exercise, using a logarithmic scale means that the increase of 7.8 dB represents several times more in intensity, specifically a ratio of 6.03 between the intensities detectable by the man compared to the woman.
Hearing Threshold
The hearing threshold is the minimum sound level of a tone that is just audible by the average human ear. This can vary greatly between individuals and even more so due to age or gender, as highlighted in the exercise. There, the woman's hearing threshold is defined relative to the man's, as she can hear the sound at a lower intensity level.

  • The woman's hearing threshold sets a baseline to measure the man's required intensity for audibility.
  • The difference in their thresholds, measured in decibels, reflects the comparative sensitivity of their hearing.
  • Age and health may affect one's hearing threshold over time.
Intensity Level Increase
The concept of intensity level increase refers to how much more intense a sound needs to be for an individual compared to a baseline. In the exercise, the man's hearing requires an intensity level increase of 7.8 dB over what the woman can hear. By understanding that decibels are logarithmic, we learn that even small-sounding numbers like 7.8 represent significant changes in intensity.

  • An increase of 7.8 dB means that the man hears the sound at an intensity 6.03 times greater than the woman.
  • Each 10 dB increase results in a perceived doubling of loudness, yet an actual ten-fold increase in intensity.
  • This illustrates the efficiency of using decibels and logarithmic scales to express and calculate everyday changes in sound intensity.

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Most popular questions from this chapter

A monatomic ideal gas \((\gamma=1.67)\) is contained within a box whose volume is \(2.5 \mathrm{m}^{3} .\) The pressure of the gas is \(3.5 \times 10^{s} \mathrm{Pa}\). The total mass of the gas is \(2.3 \mathrm{kg}\). Find the speed of sound in the gas.

A car is accelerating while its horn is sounding. Just after the car passes a stationary person, the person hears a frequency of \(966.0 \mathrm{Hz}\) Fourteen seconds later, the frequency heard by the person has decreased to \(912.0 \mathrm{Hz}\). When the car is stationary, its horn emits a sound whose frequency is \(1.00 \times 10^{3} \mathrm{Hz} .\) The speed of sound is \(343 \mathrm{m} / \mathrm{s} .\) What is the acceleration of the car?

A convertible moves toward you and then passes you; all the while, its loudspeakers are producing a sound. The speed of the car is a constant \(9.00 \mathrm{m} / \mathrm{s},\) and the speed of sound is \(343 \mathrm{m} / \mathrm{s} .\) What is the ratio of the frequency you hear while the car is approaching to the frequency you hear while the car is moving away?

Two sources of sound are located on the \(x\) axis, and each emits power uniformly in all directions. There are no reflections. One source is positioned at the origin and the other at \(x=+123 \mathrm{m}\). The source at the origin emits four times as much power as the other source. Where on the \(x\) axis are the two sounds equal in intensity? Note that there are two answers.

A steel cable has a cross-sectional area \(2.83 \times 10^{-3} \mathrm{m}^{2}\) and is kept under a tension of \(1.00 \times 10^{4} \mathrm{N}\). The density of steel is \(7860 \mathrm{kg} / \mathrm{m}^{3}\). Note that this value is not the linear density of the cable. At what speed does a transverse wave move along the cable?
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