The loudest sound measured one night during a hockey game was 112 dB. The loudest sound measured during a hockey game the next night was 118 dB. What fraction of sound intensity of the second game was the sound intensity of the first game?
[tex]L=10 \log \left(\frac{1}{t_0}\right)[/tex]
[tex]L=[/tex] loudness, in decibels
[tex]I=[/tex] sound intensity, in watts [tex]/ m ^2[/tex]
[tex]I_0=10^{-12} watts / m^2[/tex]
A. 0.25
B. 0.78
C. 0.95
D. 0.99
Answer (1)
Define L 1 and L 2 as the loudness of the first and second games, respectively, and I 1 and I 2 as their sound intensities.
Use the formula L = 10 lo g 10 ( I 0 I ) to express I 1 and I 2 in terms of L 1 and L 2 .
Find the fraction I 2 I 1 = 1 0 10 L 1 − L 2 .
Substitute the given values L 1 = 112 and L 2 = 118 to find the fraction: I 2 I 1 = 1 0 − 0.6 ≈ 0.25 .
0.25
Explanation
Understanding the Problem and Given Formula Let L 1 = 112 dB and L 2 = 118 dB be the loudness of the first and second games, respectively. Let I 1 and I 2 be the sound intensities of the first and second games, respectively. We are given the formula: L = 10 lo g 10 ( I 0 I ) where L is the loudness in decibels, I is the sound intensity in watts/ m 2 , and I 0 = 1 0 − 12 watts/ m 2 .
We have L 1 = 10 lo g 10 ( I 0 I 1 ) and L 2 = 10 lo g 10 ( I 0 I 2 ) . We want to find the fraction I 2 I 1 .
Solving for Sound Intensities From L 1 = 10 lo g 10 ( I 0 I 1 ) , we can solve for I 1 :
10 L 1 = lo g 10 ( I 0 I 1 ) 1 0 10 L 1 = I 0 I 1 I 1 = I 0 ⋅ 1 0 10 L 1 Similarly, for I 2 :
I 2 = I 0 ⋅ 1 0 10 L 2
Calculating the Fraction Now we can find the fraction I 2 I 1 :
I 2 I 1 = I 0 ⋅ 1 0 10 L 2 I 0 ⋅ 1 0 10 L 1 = 1 0 10 L 2 1 0 10 L 1 = 1 0 10 L 1 − L 2 Substitute L 1 = 112 and L 2 = 118 :
I 2 I 1 = 1 0 10 112 − 118 = 1 0 10 − 6 = 1 0 − 0.6 Using a calculator, we find that 1 0 − 0.6 ≈ 0.25118864315 .
Final Answer Therefore, the fraction of the sound intensity of the second game that was the sound intensity of the first game is approximately 0.25118864315. Rounding to two decimal places, we get 0.25.
Examples
Sound intensity, measured in decibels, is used in many fields, including acoustics, environmental science, and audio engineering. For example, understanding the relative loudness of sounds is important in designing concert halls or noise barriers. If a concert produces 120 dB at a certain location and we want to reduce the sound intensity to a level of 90 dB, we can use the formula to calculate the necessary reduction factor. This helps engineers design effective soundproofing materials and strategies to protect people from excessive noise exposure.
