The loudness, [tex]$L$[/tex], measured in decibels (Db), of a sound intensity, [tex]$I$[/tex], measured in watts per square meter, is defined as [tex]$L-10 \log \frac{1}{I_0}$[/tex], where [tex]$10-10^{-12}$[/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]?
A. 2 Db
B. 22 Db
C. 60 Db
D. 110 Db
Answer (1)
Substitute the given values into the loudness formula.
Simplify the fraction inside the logarithm.
Evaluate the logarithm.
Calculate the loudness in decibels: 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 want to find the loudness L of the rock concert in decibels.
Plugging in the values Substitute the given values of I and I 0 into the formula for loudness: L = 10 lo g 10 ( 1 0 − 12 1 0 − 1 ) .
Simplifying the fraction Simplify the argument of the logarithm: 1 0 − 12 1 0 − 1 = 1 0 − 1 − ( − 12 ) = 1 0 − 1 + 12 = 1 0 11 .
Evaluating the logarithm Evaluate the logarithm: lo g 10 ( 1 0 11 ) = 11 .
Calculating the Loudness Multiply by 10 to find the loudness in decibels: L = 10 × 11 = 110 dB . Therefore, the approximate loudness of a rock concert with a sound intensity of 1 0 − 1 watts per square meter is 110 dB.
Examples
Understanding sound intensity is crucial in various fields, such as environmental science, acoustics, and health and safety. For instance, in environmental science, measuring sound levels helps assess noise pollution in urban areas. In acoustics, it aids in designing concert halls and recording studios for optimal sound quality. In health and safety, it's essential for determining safe exposure levels to prevent hearing damage in industrial settings or at live music events. This calculation demonstrates how to quantify the loudness of sounds, ensuring appropriate measures can be taken to protect hearing and manage noise levels effectively.
