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]$L=10 \log \frac{l}{I_0}$[/tex], where [tex]$l_0=10^{-12}$[/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 the ratio of Braylee's music intensity to Jessica's music intensity: I J I B = 1 0 − 9 1 0 − 3 .
Simplify the ratio using exponent rules: 1 0 − 3 − ( − 9 ) = 1 0 6 .
Determine that Braylee's music is 1 0 6 = 1 , 000 , 000 times louder than Jessica's music.
State the final answer: 1 , 000 , 000 .
Explanation
Understanding the Problem We are given the sound intensity of Jessica's music as I J = 1 0 − 9 watts per square meter and the sound intensity of Braylee's music as I B = 1 0 − 3 watts per square meter. We want to find how many times louder Braylee's music is than Jessica's. This is equivalent to finding the ratio of their sound intensities.
Setting up the Ratio To find how many times louder Braylee's music is than Jessica's, we need to calculate the ratio of their sound intensities: I J I B .
Calculating the Ratio We have I B = 1 0 − 3 and I J = 1 0 − 9 . Therefore, the ratio is: I J I B = 1 0 − 9 1 0 − 3
Simplifying the Ratio To simplify the ratio, we use the property of exponents: a n a m = a m − n . So, we have: 1 0 − 9 1 0 − 3 = 1 0 − 3 − ( − 9 ) = 1 0 − 3 + 9 = 1 0 6 = 1 , 000 , 000
Final Answer Therefore, Braylee's music is 1,000,000 times louder than Jessica's music.
Examples
Imagine you are adjusting the volume on two different audio devices. Jessica's music is playing softly on her computer, while Braylee is listening to loud music on her headphones. The calculation we performed helps us understand the vast difference in sound intensity between the two scenarios. This concept is useful in various fields, such as audio engineering, environmental noise assessment, and workplace safety, where understanding sound intensity levels is crucial for protecting hearing and creating comfortable listening environments.
Braylee's music is 1,000,000 times louder than Jessica's. This is calculated by comparing their sound intensities, which are 1 0 − 3 and 1 0 − 9 watts per square meter respectively. The final result shows a very significant difference in loudness between the two.
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