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]$L=10 \log \frac{l}{l_0}$[/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 (1)
Substitute the given values into the formula: L = 10 lo g 1 0 − 12 1 0 − 1 .
Simplify the fraction: 1 0 − 12 1 0 − 1 = 1 0 11 .
Calculate the logarithm: lo g ( 1 0 11 ) = 11 .
Calculate the loudness: L = 10 × 11 = 110 dB. The approximate loudness of the rock concert is 110 Db.
Explanation
Understanding the Problem We are given the formula for loudness L in decibels (dB) as L = 10 lo g I 0 I , where I is the sound intensity in watts per square meter, and I 0 = 1 0 − 12 is the least intense sound a human ear can hear. We are asked to find the loudness of a rock concert with a sound intensity of I = 1 0 − 1 watts per square meter.
Substituting the Values We substitute the given values into the formula: L = 10 lo g 1 0 − 12 1 0 − 1 .
Simplifying the Fraction We simplify the fraction inside the logarithm: 1 0 − 12 1 0 − 1 = 1 0 − 1 − ( − 12 ) = 1 0 − 1 + 12 = 1 0 11 .
Calculating the Logarithm Now we take the logarithm: lo g ( 1 0 11 ) = 11 .
Calculating the Loudness Finally, we multiply by 10: L = 10 × 11 = 110 dB.
Final Answer Therefore, the approximate loudness of the rock concert is 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 or industrial areas, ensuring they comply with regulations that protect public health. Similarly, audio engineers rely on decibel measurements to optimize sound systems in concert halls, aiming for an immersive experience without causing hearing damage. By quantifying sound levels, we can create safer and more enjoyable environments for everyone.
