Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the following information for determining sound intensity. The level of sound \(\beta\) (in decibels) with an intensity \(I\) is $$\beta=10 \log _{10} \frac{I}{I_{0}}$$ where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 49 and \(50,\) find the Ievel of each sound \(\beta\). (a) \(I=10^{-4}\) watt per \(m^{2}\) (door slamming) (b) \(I=10^{-3}\) watt per \(m^{2}\) (loud car horn) (c) \(I=10^{-2}\) watt per \(m^{2}\) (siren at 30 meters)

Step-by-Step Solution

Answer

The sound levels for the given intensities are as follows: door slamming: 80 dB, loud car horn: 90 dB and siren at 30 meters: 100 dB.

Step 1: Substitute for sound intensity 'I'

Substitute the given sound intensity \(I\) for door slamming into the formula that relates sound level and intensity. So, \(\beta = 10 \times \log_{10} \frac{10^{-4}}{10^{-12}}\)

Step 2: Calculate the value

Evaluate the expression to compute the sound level for a door slamming. \(\beta = 10 \times \log_{10} 10^{8} = 80\) dB.

Step 3: Repeat the process for the other sound intensities

Repeat the process by substituting the given sound intensities for the loud car horn and the siren at 30 meters into the formula and calculate their respective sound levels. For the loud car horn, \(\beta = 10 \times \log_{10} \frac{10^{-3}}{10^{-12}} = 90\) dB and for the siren, \(\beta = 10 \times \log_{10} \frac{10^{-2}}{10^{-12}} = 100\) dB.

Key Concepts

Decibels

Decibels are a unit used to measure sound intensity, representing how loud or soft a sound is to the human ear. The decibel scale is logarithmic, which means that every 10-decibel increase represents a tenfold increase in sound intensity. This is why decibel levels can rise quickly with seemingly small increases in the source noise.

For instance:

• A quiet whisper is about 30 dB.
• Normal conversation levels are around 60 dB.
• A door slamming, as described in the exercise, reaches 80 dB.

In understanding decibels, it's crucial to get comfortable with the logarithmic nature of the scale. A sound that's 90 dB is not twice as loud as one that's 80 dB but significantly more intense. This makes the decibel scale particularly useful for comparing sounds with widely varying intensities.

Logarithmic Scale

The concept of a logarithmic scale is central to understanding decibels. Unlike linear scales, where each step between values is equal, a logarithmic scale increases exponentially. For sound, each step up on the decibel scale reflects a multiplication of intensity, not just an addition.

This logarithmic nature allows us to manage and compare the vast range of sound intensities we encounter. For example, the faintest sound humans can detect is around 0 dB, which corresponds to an intensity of \(10^{-12}\) watts per square meter. In contrast, a loud jet engine noise at 100 dB increases this intensity by a factor of \(10^{10}\), showing just how dramatically intensity escalates on this scale.

The formula provided in the exercise, \( \beta = 10 \log_{10} \left(\frac{I}{I_{0}}\right) \), effectively uses this logarithmic relationship to convert from intensity to decibels, making it mathematically manageable to express the vast differences in sound intensity levels that we can encounter every day.

Sound Levels

Sound levels help us categorize how intense a sound is, commonly measured in decibels. They are critical in understanding environmental noise and protecting against potential hearing damage. Different sounds produce different levels of decibels, as detailed in the original exercise's examples.

For instance:

• A door slamming registers about 80 dB, considered annoying but not damaging for brief exposures.
• A loud car horn, at 90 dB, could be harmful if listened to repeatedly for extended periods.
• A siren at 30 meters reaching 100 dB quickly crosses into levels that could damage hearing without protection.

By recognizing these sound levels, you can appreciate why certain environments require protective measures. When sound levels escalate to 85 dB or more, prolonged exposure increases the risk of hearing loss. Understanding these levels allows for more informed decisions about noise exposure and the necessity of hearing protection.