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 find the maximum distance at which the porpoise will still be recorded, we need to relate the intensity of the sound emitted by the porpoise to the intensity recorded by the underwater microphone.
First, we can use the relationship between intensity and distance, which states that intensity decreases with the square of the distance. Mathematically, this is represented by the inverse square law:
I = (I0 * r0^2) / r^2
Where:
- I is the intensity of the sound at distance r from the porpoise
- I0 is the intensity of the sound at a reference distance r0 (in this case, when the sound is recorded by the microphone)
- r0 is the reference distance (the distance at which the microphone is located)
- r is the distance between the porpoise and the microphone
We are given that the minimum intensity level the microphone can record is 30.9dB. Intensity is measured in watts per square meter (W/m^2), so we need to convert this level to intensity. The conversion from decibels to intensity is given by the formula:
I = 10^(dB / 10)
Substituting the given value into the equation:
I0 = 10^(30.9 / 10)
Now we can rearrange the inverse square law equation to solve for r:
r^2 = (I0 * r0^2) / I
Taking the square root of both sides:
r = sqrt((I0 * r0^2) / I)
Given that the porpoise emits sound at a rate of 0.0603J/s, we can assume this is the value of I (since the intensity is equal to the power emitted divided by the surface area of a sphere centered on the porpoise).
Substituting the values into the equation:
r = sqrt((I0 * r0^2) / 0.0603)
Since we want to find the maximum distance at which the animal will still be recorded, we can assume that the minimum intensity recorded will be at this maximum distance. Therefore, we can set the minimum intensity to I0:
30.9 = 10 * log10((I0 * r0^2) / 0.0603)
Dividing by 10 and taking the exponent:
3.09 = log10((I0 * r0^2) / 0.0603)
Using properties of logarithms, we can rewrite the equation:
10^3.09 = (I0 * r0^2) / 0.0603
Solving for I0 * r0^2:
(I0 * r0^2) = 0.0603 * 10^3.09
Now we can substitute this value back into the equation for r:
r = sqrt((0.0603 * 10^3.09) / 0.0603)
Simplifying:
r = sqrt(10^3.09)
Evaluating the square root:
r ≈ 9.844 meters
Therefore, the maximum distance at which the porpoise will still be recorded is approximately 9.844 meters.