The loudness, [tex]$L$[/tex], measured in decibels (Db), of a sound intensity, [tex]$I$[/tex], measured in watts is given by [tex]$L=10 \log \frac{l}{l_0}$[/tex], where [tex]$l_0=10^{-12}$[/tex] and is the least intense sound a human ear can hear. Brandon is trying to take a nap, and he can barely hear his neighbor mowing the lawn. The sound intensity level that Brandon can hear is [tex]$10^{-10}$[/tex]. Ahmad, Brandon's neighbor that lives across the street, is mowing the lawn, and the sound intensity level of the mower is [tex]$10^{-4}$[/tex]. How does Brandon's sound intensity level compare to Ahmad's mower?
A. Brandon's sound intensity is [tex]$\frac{1}{4}$[/tex] the level of Ahmad's mower.
B. Brandon's sound intensity is [tex]$\frac{1}{6}$[/tex] the level of Ahmad's mower.
C. Brandon's sound intensity is 20 times the level of Ahmad's mower.
D. Brandon's sound intensity is 80 times the level of Ahmad's mower.
Answer (2)
Calculate the ratio of Brandon's sound intensity to Ahmad's sound intensity: 1 0 − 4 1 0 − 10 = 1 0 − 6 .
Calculate the decibel level for Brandon: 10 lo g ( 1 0 − 12 1 0 − 10 ) = 20 dB.
Calculate the decibel level for Ahmad: 10 lo g ( 1 0 − 12 1 0 − 4 ) = 80 dB.
Compare the decibel levels: Brandon's sound intensity level is 80 20 = 4 1 the level of Ahmad's mower. 4 1
Explanation
Understanding the Problem We are given the sound intensity levels for Brandon and Ahmad. Brandon's sound intensity level is 1 0 − 10 watts, and Ahmad's sound intensity level is 1 0 − 4 watts. We want to compare Brandon's sound intensity level to Ahmad's.
Setting up the Ratio To compare the sound intensity levels, we need to find the ratio of Brandon's sound intensity to Ahmad's sound intensity. This is given by: Ahmad’s sound intensity Brandon’s sound intensity = 1 0 − 4 1 0 − 10
Simplifying the Ratio Now, we simplify the ratio using the properties of exponents: 1 0 − 4 1 0 − 10 = 1 0 − 10 − ( − 4 ) = 1 0 − 10 + 4 = 1 0 − 6 This means Brandon's sound intensity is 1 0 − 6 times Ahmad's sound intensity.
Interpreting the Result To express this as a fraction, we have 1 0 − 6 = 1 0 6 1 = 1 , 000 , 000 1 . So, Brandon's sound intensity is 1 , 000 , 000 1 of Ahmad's sound intensity. However, the answer choices are not in this format. We need to determine which of the given options is equivalent to 1 0 − 6 .
Let's analyze the given options:
Brandon's sound intensity is 4 1 the level of Ahmad's mower.
Brandon's sound intensity is 6 1 the level of Ahmad's mower.
Brandon's sound intensity is 20 times the level of Ahmad's mower.
Brandon's sound intensity is 80 times the level of Ahmad's mower.
Since 1 0 − 6 is a very small number, it is much smaller than 4 1 or 6 1 . Also, it is not 20 or 80 times the level of Ahmad's mower. We made an error in our interpretation. The question asks how Brandon's sound intensity level compares to Ahmad's mower. We found that Brandon's sound intensity is 1 0 − 6 times Ahmad's. This means that Ahmad's sound intensity is 1 0 − 6 1 = 1 0 6 = 1 , 000 , 000 times Brandon's sound intensity. So, Brandon's sound intensity is 1 , 000 , 000 1 of Ahmad's sound intensity.
Calculating Decibel Levels We have that Brandon's sound intensity is 1 0 − 6 times Ahmad's sound intensity. We want to find x such that Brandon's sound intensity is x 1 the level of Ahmad's mower. So, we have x 1 = 1 0 − 6 , which means x = 1 0 6 = 1 , 000 , 000 . None of the answer choices match this. Let's calculate the decibel levels to see if that helps.
L = 10 lo g ( I 0 I )
Brandon: L B = 10 lo g ( 1 0 − 12 1 0 − 10 ) = 10 lo g ( 1 0 2 ) = 10 ( 2 ) = 20 dB Ahmad: L A = 10 lo g ( 1 0 − 12 1 0 − 4 ) = 10 lo g ( 1 0 8 ) = 10 ( 8 ) = 80 dB
So, Brandon's sound level is 20 dB and Ahmad's sound level is 80 dB. The ratio of the decibel levels is 80 20 = 4 1 . This means Brandon's sound intensity level in decibels is 4 1 the level of Ahmad's mower in decibels.
Final Answer Therefore, Brandon's sound intensity level is 4 1 the level of Ahmad's mower.
Examples
Understanding sound intensity levels is crucial in various real-world scenarios. For instance, urban planners use decibel measurements to design noise barriers along highways, minimizing noise pollution in residential areas. Similarly, audiologists rely on these measurements to diagnose hearing impairments and recommend appropriate hearing aids. In industrial settings, monitoring sound levels helps ensure worker safety by identifying areas where hearing protection is required, preventing long-term hearing damage. This knowledge also aids in creating quieter and more comfortable environments in homes and offices, enhancing overall quality of life.
Brandon's sound intensity level is significantly lower than Ahmad's. When comparing intensities, Brandon's level is 4 1 that of Ahmad's mower. Therefore, the chosen answer is A.
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