An underwater micriphone is used to record sounds emitted by porpoises. The minimum intensity level the instrument can record is 23.6dB. Assuming a porpoise emits sound at a rate of 0.0359J/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.
To determine the maximum distance at which the porpoise will still be recorded, we need to consider the relationship between intensity, distance, and the inverse square law.
The intensity level (I) is given in decibels (dB), and it can be converted to intensity (I0) using the formula:
I = 10^(I0 / 10)
We are given that the minimum intensity level the instrument can record is 23.6 dB, so we can convert this to intensity:
I0 = 23.6 dB
I0 = 10^(23.6 / 10)
I0 = 2000.9985 (approx.)
Next, we can use the inverse square law to relate the intensity to the distance between the sound source and the microphone:
I = I0 / (4πr²)
Where:
I is the intensity at the microphone
I0 is the intensity emitted by the porpoise
r is the distance between the porpoise and the microphone
We have the value for I0 and we want to find the maximum distance, so we rearrange the equation:
r = sqrt(I0 / (4πI))
Now we can substitute the given values:
I0 = 2000.9985
I = 0.0359 J/s
Plugging in the values:
r = sqrt(2000.9985 / (4π * 0.0359))
r ≈ sqrt(13963.6434 / 0.143)
r ≈ sqrt(97606.9895)
r ≈ 312.4455 (approx.)
Therefore, the maximum distance at which the porpoise will still be recorded is approximately 312.45 meters.