Posted by Anonymous on Tuesday, December 2, 2008 at 12:43am.
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?
Responses
physics - drwls, Tuesday, December 2, 2008 at 2:14am
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.
physics - HELP - Anonymous, Tuesday, December 2, 2008 at 11:17am
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 need to use the concept of decibels (dB) and the inverse square law.
The decibel scale is a logarithmic scale used to measure the intensity or loudness of sound. The formula to convert sound intensity ratios to dB is:
dB = 10*log10(I1/I0)
where I1 is the intensity being measured and I0 is a reference intensity level (usually set at the threshold of human hearing).
In this case, we have two sound levels: 61.6 dB and 81.3 dB. To find the intensity ratio, we subtract the lower dB value from the higher dB value:
intensity ratio = 81.3 dB - 61.6 dB = 19.7 dB
Now, we can use the inverse square law to determine the distance of the speaker from each observer. The inverse square law states that the intensity of sound decreases as the square of the distance from the source increases.
The formula for the inverse square law is:
I1/I2 = (r2/r1)^2
where I1 and I2 are the intensities at distances r1 and r2 from the source, respectively.
In our case, let's assume that one observer is located at distance x from the speaker, while the other observer is located at distance (114 - x) from the speaker (since they are 114 m apart along the line connecting them).
Using the inverse square law, we can set up the following equation:
10^(19.7/10) = ((114 - x)/x)^2
To solve this equation, we can take the square root of both sides and then solve for x.
sqrt(10^(19.7/10)) = 114 - x/x
Now, we can simplify and solve for x.
sqrt(10^(19.7/10)) = 114 - x/x
10^(19.7/20) = (114 - x)/x
10^(19.7/20) * x = 114 - x
(10^(19.7/20) + 1) * x = 114
x = 114 / (10^(19.7/20) + 1)
After evaluating this expression, we can find the value of x, which represents the distance of the speaker from the first observer.
Finally, to find the distance of the speaker from the second observer, we can subtract x from 114:
Distance of the second observer = 114 - x
This will give us the distances of the speaker from each observer.