There is a band who uses 10000 W to radiate sound energy in all directions. What was the intensity of sound 10 meters from this speaker? What was the sound level in dB at a distance of 10 m? At what distance from the speaker was the sound level 100 dB?
To calculate the intensity of sound at a distance from a source, you can use the inverse square law. The inverse square law states that the intensity of sound decreases as the square of the distance from the source increases. The formula to calculate the intensity of sound is:
Intensity = Power / (4 * pi * r^2)
where Power is the power radiated by the source (in this case, 10,000 W) and r is the distance from the speaker (in this case, 10 m).
Substituting the values into the formula:
Intensity = 10,000 W / (4 * pi * (10 m)^2)
Simplifying the equation:
Intensity = 10,000 W / (4 * pi * 100 m^2)
= 10,000 W / (400 * pi m^2)
= 10,000 / 400π
≈ 7.96 W/m^2
Therefore, the intensity of sound at a distance of 10 meters from the speaker is approximately 7.96 W/m^2.
To calculate the sound level in dB at a given distance, you can use the following formula:
Sound Level = 10 * log10(Intensity / (10^-12 W/m^2))
where Intensity is the intensity of sound in W/m^2.
Substituting the value of the intensity we calculated earlier:
Sound Level = 10 * log10(7.96 W/m^2 / (10^-12 W/m^2))
Simplifying the equation:
Sound Level = 10 * log10(7.96 / 10^-12)
≈ 114.3 dB
Therefore, the sound level at a distance of 10 meters from the speaker is approximately 114.3 dB.
To calculate the distance from the speaker where the sound level is 100 dB, you can rearrange the sound level formula and solve for the distance. The formula becomes:
Distance = sqrt(Power / (4 * pi * (10^(Sound Level / 10))))
Substituting the values:
Distance = sqrt(10,000 W / (4 * pi * (10^(100 / 10))))
Simplifying the equation:
Distance = sqrt(10,000 W / (4 * pi * 10^10))
= sqrt(10,000 W / (4 * pi * 10^10))
≈ 0.0316 m
Therefore, the distance from the speaker where the sound level is approximately 100 dB is 0.0316 meters (or 3.16 cm).