Sound intensity model: [tex]L=10 \log \left(\frac{I}{I_0}\right)[/tex]
[tex]L =[/tex] loudness, in decibels [tex]( dB ) ; I =[/tex] sound intensity, in watts [tex]/ m ^2 ; I_0=10^{-12}[/tex] watts [tex]/ m ^2[/tex]
The loudness of a jack hammer is 96 dB. Its sound intensity is about ____
The loudness of a compactor is 94 dB. Its sound intensity is about ____
The sound intensity of the jack hammer is about ____ times the sound intensity of the compactor.
The loudness of a pile driver is 112 dB. About how many times the sound intensity of the jackhammer is the sound intensity of a pile driver? Round to the nearest ten.
____
Answer (1)
Calculate the difference in loudness between the pile driver and the jackhammer: 112 − 96 = 16 dB.
Divide the difference by 10: 10 16 = 1.6 .
Calculate 1 0 1.6 ≈ 39.81 .
Round to the nearest ten: The sound intensity of the pile driver is approximately 40 times the sound intensity of the jackhammer.
Explanation
Understanding the Problem We are given the sound intensity model L = 10 lo g ( I 0 I ) , where L is the loudness in decibels (dB), I is the sound intensity in watts/ m 2 , and I 0 = 1 0 − 12 watts/ m 2 . We are also given that the loudness of a jackhammer is 96 dB and its sound intensity is 0.004 watts/ m 2 . The loudness of a pile driver is 112 dB, and we want to find how many times the sound intensity of the jackhammer is the sound intensity of the pile driver.
Defining Variables Let L j and I j be the loudness and sound intensity of the jackhammer, respectively, and let L p and I p be the loudness and sound intensity of the pile driver, respectively. We are given L j = 96 dB and L p = 112 dB. We want to find the ratio I j I p .
Calculating the Ratio We have the formulas L j = 10 lo g ( I 0 I j ) and L p = 10 lo g ( I 0 I p ) . We can write the difference in loudness as:
L p − L j = 10 lo g ( I 0 I p ) − 10 lo g ( I 0 I j )
Using the logarithm property lo g ( a ) − lo g ( b ) = lo g ( b a ) , we get:
L p − L j = 10 lo g ( I j / I 0 I p / I 0 ) = 10 lo g ( I j I p )
Dividing by 10, we have:
10 L p − L j = lo g ( I j I p )
To find the ratio I j I p , we take the antilog (base 10) of both sides:
I j I p = 1 0 10 L p − L j
Substituting the given values, we have:
I j I p = 1 0 10 112 − 96 = 1 0 10 16 = 1 0 1.6
Finding the Answer Using a calculator, we find that 1 0 1.6 ≈ 39.81 . Rounding to the nearest ten, we get 40.
Conclusion Therefore, the sound intensity of the pile driver is approximately 40 times the sound intensity of the jackhammer.
Examples
Understanding sound intensity is crucial in many real-world scenarios. For example, city planners use these calculations to design noise barriers near highways or airports, ensuring that residential areas are not excessively disturbed by traffic noise. Similarly, in industrial settings, understanding sound intensity levels helps in implementing safety measures to protect workers' hearing from loud machinery. By quantifying and comparing sound levels, we can create safer and more comfortable environments.
