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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To determine the maximum distance at which the porpoise will still be recorded, we need to consider the relationship between intensity and distance.
The intensity of sound decreases as the distance from the source increases. This relationship is described by the inverse square law, which states that the intensity of sound is inversely proportional to the square of the distance from the source.
The formula for the inverse square law is:
I2 = I1 * (r1^2 / r2^2)
where:
I1 is the initial intensity at distance r1
I2 is the intensity at distance r2
In this case, the initial intensity (I1) is given as 0.0603 J/s and the minimum recording intensity (I2) is 10^(30.9/10) J/s (converted from decibels to watts/m^2).
Setting I2 equal to I1 and rearranging the formula gives:
0.0603 = 10^(30.9/10) * (r1^2 / r2^2)
Since we're looking for the maximum distance at which the porpoise will still be recorded, we can assume that the initial distance (r1) is at its maximum (infinity) and the final distance (r2) is the distance we want to find.
Rearranging the formula one more time gives:
r2 = sqrt((10^(30.9/10) * r1^2) / 0.0603)
Substituting r1 = infinity into the formula, we get:
r2 = sqrt((10^(30.9/10) * infinity^2) / 0.0603)
Since infinity squared is still infinity, we can simplify the formula to:
r2 = sqrt(infinity)
Therefore, the maximum distance at which the animal will still be recorded is infinity.