An underwater micriphone is used to record sounds emitted by porpoises. The minimum intensity level the instrument can record is 30.9dB. Assuming a porpoise emits sound at a rate of 0.0603J/s, what is the maximum distance at which the animal will still be recorded? Neglect sound absorption in water and treat the porpoise as a point sound source.
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Sure, I can help you with that!
To find the maximum distance at which the porpoise will still be recorded, we need to determine the intensity level of the sound at that distance and compare it to the minimum intensity level the instrument can record.
The intensity level of sound is given by the equation:
L = 10 * log10(I/I0)
Where L is the intensity level in decibels (dB), I is the intensity of the sound, and I0 is the reference intensity (which is typically the threshold of hearing, 10^-12 W/m^2).
In this case, the intensity level recorded by the microphone is 30.9 dB, so we can convert this into the intensity of the sound using the equation:
I/I0 = 10^(L/10)
Substituting the given value, we get:
I/I0 = 10^(30.9/10) = 12625.37
Now, we can use the inverse square law for sound to find the intensity of the sound at the maximum distance:
I = (P/(4πr^2))
Where I is the intensity of the sound, P is the power emitted by the porpoise, and r is the distance from the porpoise.
We are given that the porpoise emits sound at a rate of 0.0603 J/s, so we can substitute this value into the equation:
I = 0.0603/(4πr^2)
Now we can set the intensities equal to each other and solve for the maximum distance:
0.0603/(4πr^2) = 12625.37
Rearranging the equation and solving for r:
r^2 = 0.0603/(4π * 12625.37)
r = √(0.0603/(4π * 12625.37))
Using a calculator, we can find that r ≈ 0.00781 meters.
Therefore, the maximum distance at which the porpoise will still be recorded is approximately 0.00781 meters.