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] ?
A. -58 Db
B. -50 Db
C. 9 Db
D. 50 Db
Answer (1)
Substitute the given values into the loudness formula.
Simplify the fraction inside the logarithm.
Apply the logarithm property to simplify the expression.
Calculate the loudness: L = 50 dB. The final answer is 50 D b .
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. 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 conversation.
Substituting the 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 − 7
Simplifying the Fraction Simplify the fraction inside the logarithm: 1 0 − 12 1 0 − 7 = 1 0 − 7 − ( − 12 ) = 1 0 − 7 + 12 = 1 0 5
Substituting Back Substitute this value back into the equation for L :
L = 10 lo g 10 ( 1 0 5 )
Using Logarithm Properties Use the property of logarithms that lo g 10 ( 1 0 x ) = x :
L = 10 ⋅ 5 = 50
Final Answer Therefore, the loudness of the dinner conversation is approximately 50 dB.
Examples
Understanding sound intensity is crucial in various real-world scenarios. For instance, architects use decibel measurements to design soundproof rooms, ensuring minimal noise pollution. Similarly, environmental scientists monitor noise levels in urban areas to assess the impact on public health. By knowing how to calculate loudness, we can better appreciate and control our auditory environment, whether it's enjoying a quiet library or protecting our ears at a rock concert.
