L = loudness, in decibels $(dB); I =$ sound intensity, in watts $/ m ^2 ; I_0=10^{-12}$ watts $/ m ^2$
The loudness of a jack hammer is 96 dB. Its sound intensity is about $\square$
The loudness of a compactor is 94 dB. Its sound intensity is about $\square$
Answer (1)
Use the formula L = 10 lo g 10 ( I 0 I ) to find the sound intensity I given the loudness L and I 0 = 1 0 − 12 .
For the jack hammer with L = 96 dB, calculate I = 1 0 − 2.4 ≈ 0.00398 watts/m^2.
For the compactor with L = 94 dB, calculate I = 1 0 − 2.6 ≈ 0.00251 watts/m^2.
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 .
Explanation
Understanding the Problem We are given the loudness formula L = 10 lo g 10 ( 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 Jackhammer Intensity For the jack hammer, L = 96 dB. We substitute this into the formula and solve for I :
96 = 10 lo g 10 ( 1 0 − 12 I ) 10 96 = lo g 10 ( 1 0 − 12 I ) 9.6 = lo g 10 ( 1 0 − 12 I ) 1 0 9.6 = 1 0 − 12 I I = 1 0 9.6 × 1 0 − 12 I = 1 0 − 2.4 ≈ 0.00398 watts/m 2
Calculating Compactor Intensity For the compactor, L = 94 dB. We substitute this into the formula and solve for I :
94 = 10 lo g 10 ( 1 0 − 12 I ) 10 94 = lo g 10 ( 1 0 − 12 I ) 9.4 = lo g 10 ( 1 0 − 12 I ) 1 0 9.4 = 1 0 − 12 I I = 1 0 9.4 × 1 0 − 12 I = 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 various real-world scenarios. For instance, city planners use these calculations to assess noise pollution levels from construction equipment like jackhammers and compactors. By knowing the sound intensity, they can implement measures to protect residents from excessive noise exposure, such as erecting noise barriers or scheduling noisy activities during specific hours. This ensures a healthier and more comfortable living environment for everyone.
