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
The sound level difference is 14.8 dB. That means that the sound intensity ratio is 10^0.148 = 30.2
That means one speaker is sqrt 30.2 = 5.50 times farther way from the source than the other. If x is the shorter distance
x = 5.50 x = 6.50 x = 110 m.
x = 16.9 m The other distace is 93.1 m
Hey sorry about the 3 posts, It was an accident.
Just a quick question, why do you need to find the sound level difference?
One more thing,
how did you find 10^0.148 be 30.2? isn't it 1.406. I do see you used a ratio, but what was divided?
I meant 10^1.48 = 30.2. Sorry about the typo. The 30.2 is correct. 1.48 is 1/10 of the dB difference in received sopund power levels.
If it is a mystery where the 10^(0.1*dB difference)comes from, that is the definition of decibals in terms of intensity ratio
Log10(I2/I1) = 1/10 (dB2 - dB1)
10^[0.1(dB2-dB1)] = I2/I1
how do you get 6.50
and the final answers to be 16.9m & 93.1m?
I2/I1 = (r2)^2/(r1)^2
square root of (I2/I1) = r2/r1
This will be the difference between the two.
Therefore (distance between the two people/ square root of (I2/I1)) = smallest distance from loud speaker.
And distance between the two people minus the smallest distance is the largest distance.
To solve this problem, you will need to use the inverse square law for sound intensity. The formula is:
I1/I2 = (r2/r1)^2
Where:
I1 is the intensity measured by one observer (in this case, the observer recording a sound level of 60.1 dB),
I2 is the intensity measured by the other observer (74.9 dB),
r1 is the distance from the loudspeaker to the observer recording I1,
r2 is the distance from the loudspeaker to the observer recording I2.
Let's solve the problem step by step:
Step 1: Convert the sound levels to sound intensity values.
To convert from decibels to intensity, use the formula:
I = 10^((dB level - 120)/10)
For the observer recording 60.1 dB:
I1 = 10^((60.1 - 120)/10) = 10^-5.99 ≈ 1.02 x 10^-6
For the observer recording 74.9 dB:
I2 = 10^((74.9 - 120)/10) = 10^-4.71 ≈ 2.75 x 10^-5
Step 2: Apply the inverse square law formula:
I1/I2 = (r2/r1)^2
Substituting the known values:
1.02 x 10^-6 / 2.75 x 10^-5 = (r2/r1)^2
Step 3: Solve for (r2/r1):
(r2/r1)^2 = 1.02 x 10^-6 / 2.75 x 10^-5
Taking the square root of both sides:
r2/r1 = √(1.02 x 10^-6 / 2.75 x 10^-5)
Step 4: Calculate r1 and r2:
Let's solve for r1 first:
r1 = r2 / √(1.02 x 10^-6 / 2.75 x 10^-5)
Next, use the fact that the total distance between the observers is 110 m:
r2 + r1 = 110
So, r2 = 110 - r1
Substituting r2 into the equation we got from Step 4:
110 - r1 = r1 / √(1.02 x 10^-6 / 2.75 x 10^-5)
Now, you can solve this equation to find the values of r1 and r2.