For a speaker whose sound power output is 140 dB, the distance from the speaker at which the sound level will be reduced to 85 dB is ?
To determine the distance from the speaker at which the sound level will be reduced to 85 dB, we can use the inverse square law for sound propagation.
The inverse square law states that sound intensity decreases with the square of the distance from the source.
The formula to calculate the sound intensity (I) at a given distance (d2) from the source, given the initial sound power output (P1) at a reference distance (d1) is:
I2 = (d1/d2)^2 * I1
Let's calculate the distance (d2):
P1 = 140 dB (sound power output at d1)
I1 = 10^(P1/10) (sound intensity at d1)
P2 = 85 dB (desired sound level at d2)
I2 = 10^(P2/10) (desired sound intensity at d2)
Using the inverse square law formula, let d1 = 1 meter (reference distance):
(d1/d2)^2 = I2/I1
(1/d2)^2 = I2/I1
1/d2 = sqrt(I2/I1)
d2 = 1 / sqrt(I2/I1)
Substituting the values into the formula:
d2 = 1 / sqrt(10^(P2/10)/10^(P1/10))
d2 = 1 / sqrt(10^(85/10)/10^(140/10))
d2 = 1 / sqrt(10^8.5/10^14)
d2 = 1 / sqrt(10^(-5.5))
d2 = 1 / sqrt(0.000003162)
Calculating:
d2 ≈ 31.62 meters
Therefore, the distance from the speaker at which the sound level will be reduced to 85 dB is approximately 31.62 meters.
To find the distance from the speaker at which the sound level will be reduced to 85 dB, we need to apply the inverse square law for sound propagation.
The inverse square law states that the sound intensity decreases with the square of the distance from the source. Mathematically, it can be expressed as:
I1 / I2 = (D2 / D1)²
Where:
- I1 is the initial sound intensity (or power) at distance D1
- I2 is the final sound intensity (or power) at distance D2
In our case, we know that the initial sound power output is 140 dB, and we want to find the distance at which the sound level will be reduced to 85 dB. We can convert the decibel scale to sound intensity using the formula:
I1 = 10^((dB1 - 10*log10(I0))/10)
Where:
- dB1 is the initial sound level in decibels (140 dB in our case)
- I0 is the reference sound intensity (typically 10^(-12) W/m²)
Using this formula, we can calculate I1. Then, by rearranging the inverse square law equation, we can solve for D2:
D2 = √((I1 / I2) * D1²)
Let's calculate the distance from the speaker to the point where the sound level is reduced to 85 dB.
First, convert the initial sound level of 140 dB to sound intensity:
I1 = 10^((140 - 10*log10(10^(-12)))/10)
= 10^((140 + 120)/10)
= 10^26
= 1 * 10^26 W/m²
Now, using the inverse square law equation with I1 = 1 * 10^26 W/m², I2 = 10^((85 - 10*log10(10^(-12)))/10) = 10^(95/10) = 10^9 W/m², and D1 = 1 meter (assuming the initial distance is 1 meter from the speaker):
D2 = √((1 * 10^26) / (10^9) * (1^2))
= √(10^(26 - 9) * 1)
= √(10^17)
= 10^8 meters
Therefore, the distance from the speaker at which the sound level will be reduced to 85 dB is 100,000,000 meters or 100 kilometers.