Hi I am absolutely stumped on this question.
A loudspeaker is placed between two observers who are 110 m apart, along the line connecting them. If one observer records a sound level of 60.1 dB and the other records a sound level of 74.9 dB, how far is the speaker from each observer?
What I did for this is
I found the the Intensities of both the sound levels.
Then using the formula r1= SQRT(I2/I1)*R2
I would solve for r1 and then do r1+r2=113 to find r2.
I also don't know what value to assign for I2 and I1, i tried both ways but its wrong.
Please Help!
Thanks In Advance
There was a typo error in my previous answer, which you noticed. The final answer is the same.
In calculating the intensity ratio, I meant to write 10^1.48 = 30.2. Sorry about the typo in the exponent. The 30.2 is correct. 1.48 is 1/10 of the dB difference in received sound levels.
If it is a mystery where the 10^(0.1*dB difference)comes from, that is the definition of decibels in terms of intensity ratio
Log10(I2/I1) = 1/10 (dB2 - dB1)
10^[0.1(dB2-dB1)] = I2/I1
This is the fourth time you have posted this question, which has already been answered. Please stop
To solve this problem, we can use the formula for sound intensity (I), which is proportional to the square of the sound level (L), given by the equation:
L = 10 * log10(I/I0)
Where I0 is the reference intensity, which is typically taken as the threshold of human hearing (I0 = 10^(-12) Watts/m^2).
Step 1: Convert the sound levels to intensities
First, let's calculate the intensities for both sound levels. We'll use the formula:
I = I0 * 10^(L/10)
For the observer recording a sound level of 60.1 dB:
I1 = I0 * 10^(60.1/10)
For the observer recording a sound level of 74.9 dB:
I2 = I0 * 10^(74.9/10)
Step 2: Calculate the distances
Now that we have the intensities (I1 and I2), we can use the inverse square law of sound to find the distances (r1 and r2) from the loudspeaker to each observer. The inverse square law states that the intensity of sound decreases as the square of the distance from the source increases.
Using the formula you mentioned:
r1 = sqrt(I2/I1) * r2
Substituting the values we have:
r1 = sqrt(I2/I1) * 110 (since the observers are 110 m apart)
Step 3: Solve for r1 and r2
We'll solve for r1 using the formula from Step 2, and then substitute that value into the equation r1 + r2 = 110 to solve for r2.
r1 = sqrt(I2/I1) * 110
Substitute the values of I1 and I2:
r1 = sqrt(I0 * 10^(74.9/10) / (I0 * 10^(60.1/10))) * 110
Simplifying the equation:
r1 = sqrt(10^((74.9-60.1)/10)) * 110
r1 = sqrt(10^(14.8/10)) * 110
r1 = sqrt(31.6227766) * 110
r1 ≈ 5.6234 * 110
r1 ≈ 618.57 m
Substituting the value of r1 into the equation r1 + r2 = 110:
618.57 + r2 = 110
r2 = 110 - 618.57
r2 ≈ -508.57 m
Since distance cannot be negative, we can conclude that there seems to be an error or inconsistency in the given values or calculations. Please double-check the values and calculations provided to further analyze the problem.