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, distance, and power.
The intensity of a sound wave emitted by a point source decreases with distance according to the inverse square law. This means that the intensity is inversely proportional to the square of the distance from the source.
The formula to calculate intensity (I) is:
I = P / (4πr²)
Where:
- I is the intensity of the sound wave in watts per square meter (W/m²)
- P is the power of the sound source in watts (W)
- r is the distance from the source in meters (m)
- π is a mathematical constant (approximately 3.14159)
In this case, we are given the power (0.0603J/s) and the minimum detectable intensity level (30.9dB), which we need to convert to watts per square meter.
To convert decibels (dB) to intensity in watts per square meter (W/m²), we use the formula:
I2 = I1 * 10^(dB1/10)
Where:
- I1 is the initial intensity in W/m²
- I2 is the final intensity in W/m²
- dB1 is the initial intensity level in decibels
First, let's convert the minimum detectable intensity level from decibels to watts per square meter:
I1 = 10^(dB1/10)
I1 = 10^(30.9/10)
I1 = 10^3.09
I1 ≈ 1101.98 W/m²
Now that we have the initial intensity, we can rearrange the equation to solve for the maximum distance (r).
r = sqrt(P / (4πI))
r = sqrt(0.0603 / (4 * 3.14159 * 1101.98))
r ≈ sqrt(0.0603 / 13804.03)
r ≈ sqrt(4.373 * 10^(-6))
r ≈ 0.002092 m
Therefore, the maximum distance at which the porpoise will still be recorded is approximately 0.002092 meters or 2.092 millimeters.