A loudspeaker is placed between two observers who are 114 m apart, along the line connecting them. If one observer records a sound level of 61.6 dB and the other records a sound level of 81.3 dB, how far is the speaker from each observer?
Get the intensity ratio from the decibel difference, and apply the inverse square law so that the ratio of distances squared are the inverse ratio of the intensities.
intensity ratio = 19.7dB
I=1/r^2
r^2 = 1/19.7
r = .225m
this doesn't make sense
what am i doing wrong?
To solve this problem, we can use the inverse square law for sound intensity, which states that the intensity of sound decreases with the square of the distance from the source.
Let's assume that the distance from the loudspeaker to the first observer is x, and the distance from the loudspeaker to the second observer is 114 - x (since they are 114 m apart).
The sound levels (measured in decibels) follow a logarithmic scale, where the difference in sound level is related to the ratio of sound intensities:
(dB2 - dB1) = 10 * log10(I1 / I2)
Given that one observer records a sound level of 61.6 dB and the other records a sound level of 81.3 dB, we can plug these values into the equation:
81.3 - 61.6 = 10 * log10(I1 / I2)
Applying the logarithm and simplifying:
19.7 = 10 * log10(I1 / I2)
1.97 = log10(I1 / I2)
10^1.97 = (I1 / I2)
Now, we can use the inverse square law to relate the sound intensities with the distances:
(I1 / I2) = (dist2 / dist1)^2
Substituting the distances:
10^1.97 = ((114 - x) / x)^2
Simplifying:
10^1.97 = ((114 - x)^2 / x^2)
Taking the square root of both sides:
(10^1.97)^0.5 = (114 - x) / x
Simplifying further:
3.179 = (114 - x) / x
Cross-multiplying:
3.179x = 114 - x
Adding x to both sides:
4.179x = 114
Dividing by 4.179:
x = 27.279 m
Therefore, the distance from the loudspeaker to the first observer is approximately 27.279 m, and the distance from the loudspeaker to the second observer is approximately 114 - 27.279 = 86.721 m.