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]$L=10 \log \frac{1}{f_0}$[/tex], where [tex]$f_0=10^{-12}$[/tex] and is the least intense sound a human ear can hear. What is the approximate loudness of a dinner conversation with a sound intensity of [tex]$10^{-7}$[/tex]?
Answer (2)
Substitute the given values into the loudness formula: L = 10 lo g 10 1 0 − 12 1 0 − 7 .
Simplify the fraction: 1 0 − 12 1 0 − 7 = 1 0 5 .
Evaluate the logarithm: lo g 10 ( 1 0 5 ) = 5 .
Calculate the loudness: L = 10 × 5 = 50 dB. The approximate loudness of the dinner conversation is 50
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 dinner conversation is I = 1 0 − 7 watts per square meter. We want to find the loudness L of this dinner conversation.
Substituting 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 − 7 Now, we simplify the fraction inside the logarithm.
Simplifying the Fraction Using the properties of exponents, we have: 1 0 − 12 1 0 − 7 = 1 0 − 7 − ( − 12 ) = 1 0 − 7 + 12 = 1 0 5 So, the expression becomes: L = 10 lo g 10 ( 1 0 5 ) Now, we evaluate the logarithm.
Evaluating the Logarithm Since lo g 10 ( 1 0 5 ) = 5 , we have: L = 10 × 5 = 50 Therefore, the loudness of the dinner conversation is 50 dB.
Final Answer The loudness of a dinner conversation with a sound intensity of 1 0 − 7 watts per square meter is approximately 50 dB.
Examples
Understanding sound intensity is crucial in many real-world scenarios. For instance, architects use decibel measurements to design soundproof rooms, ensuring minimal noise pollution. Similarly, audio engineers rely on these calculations to optimize sound systems in concert halls, delivering the best possible listening experience. City planners also consider sound levels when designing residential areas near airports or highways to mitigate noise pollution and improve the quality of life for residents. The formula L = 10 lo g 10 I 0 I helps professionals quantify and manage sound levels effectively.
The loudness of a dinner conversation with a sound intensity of 1 0 − 7 watts per square meter is approximately 50 dB. This is calculated using the formula for loudness involving the reference sound intensity. The calculations show that this level is typical for normal conversations.
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