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
A. 0.0096
B. 0.004
C. 0.0021
Answer (2)
The sound intensity of a jackhammer with a loudness of 96 dB can be calculated using the formula L = 10 lo g ( I 0 I ) . After performing the calculations, the sound intensity is approximately 0.004 watts/m², which corresponds to option B. Therefore, the final answer is 0.004 watts/m².
;
Substitute the given values into the sound intensity model.
Simplify the equation to isolate the sound intensity, I .
Calculate the value of I using the properties of logarithms and exponents.
The sound intensity of the jackhammer is approximately 0.004 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 are given that the loudness of a jack hammer is 96 dB, so L = 96 . We want to find the sound intensity I .
Substituting Values Substitute L = 96 and I 0 = 1 0 − 12 into the sound intensity model equation: 96 = 10 lo g ( 1 0 − 12 I )
Dividing by 10 Divide both sides of the equation by 10: 9.6 = lo g ( 1 0 − 12 I )
Converting to Exponential Form Rewrite the equation in exponential form: 1 0 9.6 = 1 0 − 12 I
Isolating I Multiply both sides of the equation by 1 0 − 12 : I = 1 0 9.6 ⋅ 1 0 − 12
Simplifying the Exponent Simplify the exponent: I = 1 0 9.6 − 12 = 1 0 − 2.4
Calculating Sound Intensity Calculate the value of 1 0 − 2.4 to find the sound intensity I . The result of the operation is approximately 0.003981.
Final Answer Express the result in decimal form. The sound intensity I is approximately 0.003981 watts/m^2. The closest answer choice is 0.004.
Examples
Understanding sound intensity is crucial in various real-world scenarios. For instance, city planners use sound intensity models to assess noise pollution levels from traffic or construction, ensuring that residential areas remain within acceptable noise limits. Similarly, audio engineers rely on these models to design concert halls and recording studios, optimizing sound quality and minimizing unwanted reflections. Moreover, occupational health and safety professionals employ sound intensity measurements to protect workers from hearing damage in noisy industrial environments, implementing measures like noise barriers or hearing protection programs.
