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]$L=10 \log \frac{I}{I_0}$[/tex], where [tex]$I_0=10^{-12}$[/tex] and is the least intense sound a human ear can hear. Jessica is listening to soft music at a sound intensity level of [tex]$10^{-9}$[/tex] on her computer while she does her homework. Braylee is completing her homework while listening to very loud music at a sound intensity level of [tex]$10^{-3}$[/tex] on her headphones. How many times louder is Braylee's music than Jessica's?
A. [tex]$\frac{1}{3}$[/tex] times louder
B. 3 times louder
C. 30 times louder
D. 90 times louder
Answer (2)
Calculate Jessica's loudness: L J = 10 lo g 1 0 − 12 1 0 − 9 = 30 dB.
Calculate Braylee's loudness: L B = 10 lo g 1 0 − 12 1 0 − 3 = 90 dB.
Find the ratio of Braylee's loudness to Jessica's loudness: L J L B = 30 90 = 3 .
Braylee's music is 3 times louder than Jessica's. $\boxed{3}
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 reference intensity. We are given Jessica's music intensity I J = 1 0 − 9 and Braylee's music intensity I B = 1 0 − 3 . We want to find how many times louder Braylee's music is than Jessica's.
Calculating Jessica's Loudness First, we calculate the loudness of Jessica's music, L J , using the formula: L J = 10 lo g I 0 I J = 10 lo g 1 0 − 12 1 0 − 9 = 10 lo g ( 1 0 − 9 − ( − 12 ) ) = 10 lo g ( 1 0 3 ) = 10 ⋅ 3 = 30 dB .
Calculating Braylee's Loudness Next, we calculate the loudness of Braylee's music, L B , using the formula: L B = 10 lo g I 0 I B = 10 lo g 1 0 − 12 1 0 − 3 = 10 lo g ( 1 0 − 3 − ( − 12 ) ) = 10 lo g ( 1 0 9 ) = 10 ⋅ 9 = 90 dB .
Finding the Ratio Now, we find the ratio of Braylee's loudness to Jessica's loudness: L J L B = 30 90 = 3.
Examples
Understanding sound intensity is crucial in various fields, such as acoustics, environmental science, and audio engineering. For instance, consider designing a concert hall where the sound level needs to be consistent throughout the venue. By calculating the loudness at different points using the sound intensity formula, engineers can optimize the hall's design to ensure an enjoyable listening experience for everyone. This involves managing sound reflections, absorption, and diffusion to achieve the desired sound levels and clarity.
Braylee's loudness level is calculated as 90 dB, and Jessica's as 30 dB. The ratio of their loudness levels shows that Braylee's music is 3 times louder than Jessica's. Therefore, the answer is B. 3 times louder.
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