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.
Determine the ratio of intensities: I J I B = 1 0 − 9 1 0 − 3 = 1 0 6 .
Braylee's music is 1 0 6 times more intense than Jessica's, but none of the options match this result. There may be an error in the question or the options.
Explanation
Understanding the Problem We are given the formula for loudness in decibels: L = 10 lo g I 0 I , where I is the sound intensity and I 0 = 1 0 − 12 . 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. This can be found by comparing the ratio of their intensities.
Calculating Loudness First, let's calculate the loudness of Jessica's music: L 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 Next, let's calculate the loudness of Braylee's music: L 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 of Intensities To find how many times louder Braylee's music is than Jessica's, we can compare the ratio of their intensities: I J I B = 1 0 − 9 1 0 − 3 = 1 0 − 3 − ( − 9 ) = 1 0 − 3 + 9 = 1 0 6 = 1 , 000 , 000 This means Braylee's music is 1,000,000 times more intense than Jessica's music. However, the question asks how many times louder it is, not how many times more intense. The difference in loudness is L B − L J = 90 − 30 = 60 dB. Each 10 dB increase represents a factor of 10 in intensity. So a 60 dB difference represents 1 0 6 times the intensity.
Determining How Many Times Louder Alternatively, we can directly calculate the ratio of the intensities: I J I B = 1 0 − 9 1 0 − 3 = 1 0 − 3 − ( − 9 ) = 1 0 6 = 1 , 000 , 000 So, Braylee's music is 1,000,000 times more intense than Jessica's music. The question asks how many times louder Braylee's music is than Jessica's. Since the loudness is proportional to the logarithm of the intensity, we can say that Braylee's music is 1 0 6 times louder than Jessica's. However, the options provided do not match this result. Let's re-examine the question. The question is asking how many times louder Braylee's music is than Jessica's. This is equivalent to asking for the ratio of their intensities, which we have already calculated as 1 0 6 . However, the options are much smaller. It seems there might be a misunderstanding of the term 'times louder'. The correct interpretation is the ratio of intensities, which is 1 0 6 . However, none of the options match this. Let's look at the difference in decibels: 90 − 30 = 60 . If we divide this by 10, we get 6. Then 1 0 6 = 1 , 000 , 000 . This is still not among the options. Let's reconsider the question. The question is asking how many times louder Braylee's music is than Jessica's. This is the ratio of the intensities: 1 0 − 9 1 0 − 3 = 1 0 6 . The options are 3 1 , 3, 30, 90. None of these are 1 0 6 . There must be an error in the question or the options. However, if the question was asking for the ratio of the loudness in decibels divided by 2, then the answer would be 30 90 = 3 . But this is not the correct interpretation. The correct interpretation is the ratio of intensities, which is 1 0 6 .
Final Answer and Conclusion The ratio of the intensities of Braylee's music to Jessica's music is: I J I B = 1 0 − 9 1 0 − 3 = 1 0 6 = 1 , 000 , 000 Since none of the given options match this result, there might be an error in the question or the provided options. However, based on the given information and the formula for loudness, the correct answer should be 1,000,000. Since this is not an option, we must assume that the question is flawed. However, if we were to choose the closest answer from the given options, we would consider the ratio of the decibel levels: 30 90 = 3 . This is one of the options. However, this is not the correct way to interpret the question. The correct way is to find the ratio of the intensities, which is 1 0 6 .
Examples
Understanding sound intensity is crucial in various fields. For instance, in environmental science, it helps in assessing noise pollution levels near airports or industrial areas. In audio engineering, it's used to design sound systems that deliver optimal listening experiences without causing hearing damage. Moreover, in healthcare, audiologists use sound intensity measurements to diagnose hearing impairments and recommend appropriate treatments. By quantifying sound levels, we can create safer and more comfortable environments for everyone.
Braylee's music is much louder than Jessica's, with loudness levels of 30 dB and 90 dB respectively. While the actual intensity ratio suggests Braylee's music is 1,000,000 times more intense, the choice aligning most closely with perceived loudness based on decibel level differences is B) 3 times louder. The given calculations and results lead to this interpretation within the options provided.
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