Problem 8 - A Graphical Approach to College Algebra

Question

Sound Intensity Use the formula $$d=10 \log \frac{I}{I_{0}}$$ to estimate the average decibel level for each sound with the given intensity \(I .\) For comparison, conversational speech has a sound level of about 60 decibels. (a) Jackhammer: \(31,620,000,000 I_{0}\). (b) iPhone 5 speakers: \(10^{1 /} l_{0}\). (c) Rock singer screaming into microphone: \(10^{14} I_{0}\).

Step-by-Step Solution

Answer

(a) 105 decibels, (b) 10 decibels, (c) 140 decibels.

Step 1: Understanding the Formula

The sound intensity formula is given by \( d = 10 \log \frac{I}{I_{0}} \), where \( d \) is the decibel level, \( I \) is the intensity of the sound, and \( I_{0} \) is the reference intensity. The logarithmic operation indicates how many times more intense a sound is compared to the reference level.

Step 2: Calculating Decibels for Jackhammer

For the jackhammer, the intensity \( I \) is \( 31,620,000,000 I_{0} \). Substitute into the formula: \[ d = 10 \log \frac{31,620,000,000 I_{0}}{I_{0}} = 10 \log 31,620,000,000 \]Finally, calculate the logarithm: \[ d \approx 10 \times 10.5 = 105 \text{ decibels} \]

Step 3: Calculating Decibels for iPhone 5 Speakers

For the iPhone 5 speakers, the intensity \( I \) is \( 10^{1} I_{0} \). Substitute into the formula: \[ d = 10 \log \frac{10^{1} I_{0}}{I_{0}} = 10 \log 10^{1} \]Since the logarithm of 10 is 1:\[ d = 10 \times 1 = 10 \text{ decibels} \]

Step 4: Calculating Decibels for Rock Singer

For the rock singer, the intensity \( I \) is \( 10^{14} I_{0} \). Substitute into the formula: \[ d = 10 \log \frac{10^{14} I_{0}}{I_{0}} = 10 \log 10^{14} \]Since the logarithm of \(10^{14}\) is 14:\[ d = 10 \times 14 = 140 \text{ decibels} \]

Key Concepts

Decibel Level

Decibel level is a unit used to measure the intensity of sound. It reflects how loud or soft a sound is perceived by the human ear. The term "decibel" is derived from the word "bel," named after Alexander Graham Bell, the inventor of the telephone. One decibel (dB) is one-tenth of a bel and provides a more convenient scale for representing sound intensity.
The decibel scale is logarithmic, which means it measures sound in multiples of ten. This makes it easier to express very large or very small numbers. For example, a sound that is 10 times more intense is perceived as only twice as vibrant or loud.
Understanding decibel levels helps us gauge whether a sound is safe or potentially harmful to human hearing. Normal conversation typically occurs around 60 decibels, while noises above 85 decibels can lead to hearing damage over time. Therefore, it's crucial to monitor exposure, especially in environments with loud noises like concerts or construction sites.

Logarithmic Scale

Logarithmic scales are used when we want to describe exponential growth or decay, like in sound intensity. Unlike a linear scale, where each unit increment represents a constant amount, in a logarithmic scale each unit step up corresponds to a tenfold increase (or decrease) in the quantity measured.
The formula for calculating sound intensity in decibels: \[d = 10 \log \frac{I}{I_0} \]This formula illustrates how logarithmic scaling works. It simplifies the comparison of vastly different quantities, transforming multiplicative changes into additive changes.
This means on a logarithmic scale, an intensity ratio of 10:1 is represented as an increase of 10 decibels, while a ratio of 100:1 represents a 20 decibel increase. Such scaling makes it easier to graphically display and interpret data over wide ranges of values, keeping numbers manageable.
Logarithmic scales are crucial in many fields, such as seismology and electronics, in addition to acoustics, because they can accurately represent phenomena that span several orders of magnitude.

Reference Intensity

Reference intensity (\(I_0 \)) is a standard or baseline level of intensity used in calculations to compare other sounds. It is defined as the threshold of hearing, or the quietest sound that can be detected by the average human ear under ideal conditions. \[I_0 = 10^{-12} \text{ watts/m}^2\]This reference point remains consistent across measurements, so different sounds can be compared using the same scale.
For example, when calculating the decibel level of a rock singer or a jackhammer, the reference intensity helps maintain uniformity in calculations. It allows the formula to relate sound intensity directly to human perception of loudness.
Without a reference intensity, decibel calculations would be less accurate and more variable, complicating the understanding of sound levels across different contexts. This standardization crucially aids in hearing conservation and noise regulation.