Problem 117
Question
The loudness level of a sound, D, in decibels, is given by the formula $$ D=10 \log \left(10^{12} I\right) $$ where I is the intensity of the sound, in watts per meter2. Decibel levels range from 0, a barely audible sound, to 160, a sound resulting in a ruptured eardrum. (Any exposure to sounds of 130 decibels or higher puts a person at immediate risk for hearing damage.) Use the formula to solve Exercises 117–118. The sound of a blue whale can be heard 500 miles away, reaching an intensity of \(6.3 \times 10^{6}\) watts per meter? Determine the decibel level of this sound. At close range, can the sound of a blue whale rupture the human eardrum?
Step-by-Step Solution
Verified Answer
The decibel level of the sound produced by a blue whale, given the intensity of \( 6.3 \times 10^{6} \) watts per meter squared, can be determined by solving the formula \( D=10 \log \left(10^{12} I\right) \). If the calculated level is 160 or more, then the sound can rupture a human eardrum.
1Step 1: Identify the known and unknown
The formula for determining the loudness level of a sound is \( D=10 \log \left(10^{12} I\right) \). The provided information is that the intensity, \( I \), from the sound of a blue whale is \( 6.3 \times 10^{6} \) watts per meter squared. The goal is to find \( D \), the decibel level.
2Step 2: Substitute values into the equation
Substitute \( I = 6.3 \times 10^{6} \) into the formula to get: \( D=10 \log \left(10^{12} \times 6.3 \times 10^{6}\right) \)
3Step 3: Simplify the equation
Simplify the formula, specifically the part within the logarithm, to get: \( D=10 \log \left(6.3 \times 10^{18}\right) \).
4Step 4: Solve the equation
Evaluate \( \log \left(6.3 \times 10^{18}\right) \) and multiply the result by 10 to get the decibel level \( D \).
5Step 5: Comparison with danger level
Compare the calculated decibel level with the known danger level of 160. If the calculated level is 160 or more, then the sound of a blue whale can rupture a human eardrum.
Key Concepts
Decibel Level CalculationLogarithmic FunctionsIntensity of Sound
Decibel Level Calculation
Sound intensity levels are often measured in decibels (dB), which is a unit of measurement that quantifies sound pressure level relative to the faintest sound audible to the human ear. The given formula for calculating the decibel level of a sound is:
\[ D = 10 \log(10^{12} I) \]
where \( D \) represents the decibel level, and \( I \) is the intensity of the sound in watts per square meter (W/m²). When using this formula, it is essential to use the intensity value in the correct units to ensure an accurate calculation of the decibel level.
For instance, with the sound of a blue whale at \( 6.3 \times 10^{6} \) W/m², you would first multiply this intensity by \( 10^{12} \) to scale the sound intensity relative to the threshold of human hearing. Next, apply the logarithmic function to this result, and finally, multiply by 10 to find the decibel level. Through this calculation, if the resulting dB level is above 130, it is considered harmful to human ears, and levels of 160 dB or more could potentially rupture a human eardrum.
\[ D = 10 \log(10^{12} I) \]
where \( D \) represents the decibel level, and \( I \) is the intensity of the sound in watts per square meter (W/m²). When using this formula, it is essential to use the intensity value in the correct units to ensure an accurate calculation of the decibel level.
For instance, with the sound of a blue whale at \( 6.3 \times 10^{6} \) W/m², you would first multiply this intensity by \( 10^{12} \) to scale the sound intensity relative to the threshold of human hearing. Next, apply the logarithmic function to this result, and finally, multiply by 10 to find the decibel level. Through this calculation, if the resulting dB level is above 130, it is considered harmful to human ears, and levels of 160 dB or more could potentially rupture a human eardrum.
Logarithmic Functions
Logarithmic functions are a type of mathematical operation essential to various scientific calculations, including decibel level computation. The logarithm, denoted as \( \log \), represents the exponent to which a base must be raised to produce a given number. In the formula \( D=10 \log(10^{12}I) \), the base is 10, which is commonly used in scientific contexts and is referred to as the common logarithm.
Using a logarithm in calculating sound intensity levels compresses the wide range of sound powers into a smaller, more manageable scale. This is due to the logarithm's property of transforming multiplicative relationships into additive ones, which explains why doubling the sound intensity increases the dB level by a fixed amount instead of doubling it. Understanding how to simplify and calculate logarithm expressions is vital for accurately determining decibel levels from intensity measurements. Step by step, one would take the log of the product of intensity and the reference intensity, then apply any constants, in this case, multiplying by 10 to get the final decibel level.
Using a logarithm in calculating sound intensity levels compresses the wide range of sound powers into a smaller, more manageable scale. This is due to the logarithm's property of transforming multiplicative relationships into additive ones, which explains why doubling the sound intensity increases the dB level by a fixed amount instead of doubling it. Understanding how to simplify and calculate logarithm expressions is vital for accurately determining decibel levels from intensity measurements. Step by step, one would take the log of the product of intensity and the reference intensity, then apply any constants, in this case, multiplying by 10 to get the final decibel level.
Intensity of Sound
Sound intensity, denoted as \( I \), is a measure of the power per unit area carried by a sound wave. It is expressed in watts per square meter (W/m²). In simpler terms, it's a way to describe how much energy is spread over a certain area, akin to how bright a light is depending on how much light energy is hitting a surface.
In our context, a blue whale's sound is extremely powerful, yet if you are far enough away, the intensity decreases, which is partly why it does not rupture our eardrums at long distances. The intensity of sound is inversely proportional to the square of the distance from the source, following the inverse square law. So, as you double your distance from the sound source, the sound intensity falls to a quarter of its original value. This is a crucial feature when considering the safe exposure levels of any sound source, especially in situations where the original intensity is as massive as those produced by the calls of the blue whale.
In our context, a blue whale's sound is extremely powerful, yet if you are far enough away, the intensity decreases, which is partly why it does not rupture our eardrums at long distances. The intensity of sound is inversely proportional to the square of the distance from the source, following the inverse square law. So, as you double your distance from the sound source, the sound intensity falls to a quarter of its original value. This is a crucial feature when considering the safe exposure levels of any sound source, especially in situations where the original intensity is as massive as those produced by the calls of the blue whale.
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