The loudness, [tex]$L$[/tex], measured in decibels (Db), of a sound intensity, I, measured in watts per square meter, is defined as [tex]$L =10 \log \frac{ l }{ l _0}$[/tex], where [tex]$l _0=10^{-12}$[/tex] and is the least intense sound a human ear can hear. What is the approximate loudness of a rock concert with a sound intensity of [tex]$10^{-1}$[/tex]?
Answer (2)
The problem gives us a formula to calculate loudness L based on sound intensity I . We substitute the given values into the formula, simplify the expression, and calculate the loudness.
Substitute I = 1 0 − 1 and I 0 = 1 0 − 12 into the formula: L = 10 lo g 10 ( 1 0 − 12 1 0 − 1 ) .
Simplify the fraction: 1 0 − 12 1 0 − 1 = 1 0 11 .
Calculate the logarithm: lo g 10 ( 1 0 11 ) = 11 .
Multiply by 10: L = 10 × 11 = 110 Db.
Explanation
Understanding the Problem We are given the formula for loudness L in decibels (dB) as:
L = 10 lo g 10 ( I 0 I )
where I is the sound intensity in watts per square meter, and I 0 = 1 0 − 12 watts per square meter is the reference intensity (the least intense sound a human ear can hear).
We are given that the sound intensity of a rock concert is I = 1 0 − 1 watts per square meter. We need to find the loudness L of the rock concert in dB.
Substituting Values Substitute the given values of I and I 0 into the formula for L :
L = 10 lo g 10 ( 1 0 − 12 1 0 − 1 )
Simplify the fraction inside the logarithm:
1 0 − 12 1 0 − 1 = 1 0 − 1 − ( − 12 ) = 1 0 − 1 + 12 = 1 0 11
So we have:
L = 10 lo g 10 ( 1 0 11 )
Calculating the Logarithm Now, we calculate the logarithm:
lo g 10 ( 1 0 11 ) = 11
This is because the logarithm base 10 of 10 raised to any power is simply that power.
Finding the Loudness Finally, multiply the result by 10 to find the loudness L :
L = 10 × 11 = 110
Therefore, the loudness of the rock concert is approximately 110 dB.
Examples
Understanding sound intensity is crucial in many real-world scenarios. For instance, city planners use decibel measurements to assess noise pollution levels from traffic and construction, ensuring they comply with regulations that protect residents' hearing. Similarly, audio engineers rely on decibel levels to optimize sound systems in concert halls and theaters, providing audiences with the best possible listening experience while preventing hearing damage. This knowledge is also vital in occupational safety, where employers must monitor noise levels in factories and provide hearing protection to workers exposed to loud machinery.
The loudness of a rock concert with a sound intensity of 1 0 − 1 watts per square meter is calculated to be approximately 110 dB using the formula L = 10 lo g 10 ( I 0 I ) , where I 0 = 1 0 − 12 watts per square meter. By substituting the values and simplifying, we find that L = 110 dB. Thus, this loudness value indicates a very loud sound environment typical of a rock concert.
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