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[/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 ______.
Answer (1)
Substitute the loudness of the jack hammer (96 dB) into the sound intensity model and solve for I , resulting in I = 1 0 − 2.4 watts/ m 2 .
Substitute the loudness of the compactor (94 dB) into the sound intensity model and solve for I , resulting in I = 1 0 − 2.6 watts/ m 2 .
The sound intensity of the jack hammer is approximately 0.00398 watts/ m 2 .
The sound intensity of the compactor is approximately 0.00251 watts/ m 2 .
0.00398 watts / m 2 , 0.00251 watts / m 2
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 need to find the sound intensity for a jack hammer with a loudness of 96 dB and a compactor with a loudness of 94 dB.
Calculating Jack Hammer Intensity For the jack hammer, we have L = 96 dB. We substitute this into the formula and solve for I :
96 = 10 lo g ( 1 0 − 12 I ) Divide both sides by 10: 9.6 = lo g ( 1 0 − 12 I ) Now, we take the antilog (base 10) of both sides: 1 0 9.6 = 1 0 − 12 I Multiply both sides by 1 0 − 12 :
I = 1 0 9.6 × 1 0 − 12 = 1 0 − 2.4 ≈ 0.00398 watts / m 2
Calculating Compactor Intensity For the compactor, we have L = 94 dB. We substitute this into the formula and solve for I :
94 = 10 lo g ( 1 0 − 12 I ) Divide both sides by 10: 9.4 = lo g ( 1 0 − 12 I ) Now, we take the antilog (base 10) of both sides: 1 0 9.4 = 1 0 − 12 I Multiply both sides by 1 0 − 12 :
I = 1 0 9.4 × 1 0 − 12 = 1 0 − 2.6 ≈ 0.00251 watts / m 2
Final Answer Therefore, the sound intensity of the jack hammer is approximately 0.00398 watts/ m 2 , and the sound intensity of the compactor is approximately 0.00251 watts/ m 2 .
Examples
Understanding sound intensity is crucial in many real-world scenarios. For example, city planners use sound intensity models to assess noise pollution levels from construction sites or traffic. By measuring the decibel levels and calculating the sound intensity, they can determine the impact on nearby residents and implement noise reduction strategies. This ensures a healthier and more comfortable living environment for everyone.
