The intensity of sound from a source is measured at two points along a line from the source. The points are separated by 14.7 m, the sound level is 67.10 dB at the first point and 55.68 dB at the second point. How far is the source from the first point?

To find the distance between the sound source and the first point, we can make use of the inverse square law for sound intensity.

The inverse square law states that the intensity of sound at a given point is inversely proportional to the square of the distance from the sound source. Mathematically, it can be written as:

I1/I2 = (r2/r1)^2

Where I1 and I2 are the intensities of sound at the first and second points respectively, and r1 and r2 are the distances of the first and second points from the sound source.

In this case, we are given the intensities I1 = 67.10 dB and I2 = 55.68 dB, and we need to find the distance r1.

Converting the decibel (dB) values to intensity:

I1 = 10^(67.10/10)
I2 = 10^(55.68/10)

Rearranging the formula, we have:

r1^2 = (I2/I1) * r2^2
r1 = sqrt((I2/I1) * r2^2)

Now we can substitute the known values to calculate the distance r1:

r1 = sqrt((10^(55.68/10)) / (10^(67.10/10)) * (14.7^2))

Evaluating this expression, we can find the distance r1.

To determine the distance between the source and the first point, we can use the inverse square law for sound intensity. The formula is:

I₂ = I₁ × (r₁ / r₂)²

Where:
I₁ = intensity at the first point
I₂ = intensity at the second point
r₁ = distance from the source to the first point
r₂ = distance from the source to the second point

Let's plug in the given values:

I₁ = (10^(67.10/10)) × 10^(-12) W/m² (converting dB to W/m²)
I₂ = (10^(55.68/10)) × 10^(-12) W/m²
r₁ = ?
r₂ = 14.7 m (given)

First, calculate the intensities:
I₁ = 10^(67.10/10) × 10^(-12)
I₂ = 10^(55.68/10) × 10^(-12)

Now, rearrange the formula to solve for r₁:

r₁ = √[(I₁ / I₂) × (r₂)²]

Substituting the values:

r₁ = √[(I₁ / I₂) × (14.7)²]

Calculate the square root term:

√[(I₁ / I₂) × (14.7)²] = √[(I₁ / I₂) × 216.09]

Now, substitute the calculated values for I₁ and I₂:

r₁ = √[(10^(67.10/10) × 10^(-12)) / (10^(55.68/10) × 10^(-12)) × 216.09]

Using a calculator, simplify:

r₁ ≈ √[18.5371]

r₁ ≈ 4.3085 m

Therefore, the source is approximately 4.3085 meters away from the first point.

The difference in sound intensity between the two points is 11.42 dB

A 10 dB drop corresponds to a factor of 10 dropoff in intensity (sound power per area), and 11.42 corresponds to a factor of 10^1.142 = 13.87.

Intensity obeys an inverse square law. If R1 is the distance from the source to the place where the higher signal is recorded,

P2/P1 = 1/13.87 = (R1/R2)^2
= [R1/(R1+14.7)]^2

1+ (14.7/R1) = sqrt13.87 = 3.72
14.7/R1 = 2.72

Solve for R1