What is the distance from the source to the closest observer?

[dsimpson]

Homework Statement

The intensity of the sound from a certain source is measured by two observers located at different positions along a line from the source. The observers are located on the same side of the source and are separated by 77.7 m. The observer that is closest to the source hears the sound with an intensity of 60.8 dB. The intensity of the sound heard by the more distant observer is 58.9 dB.

What is the distance from the source to the closest observer?

Homework Equations

We were given no equations but a classmate came up with the equation r1=77.7/(10^((B1-B2)/20)-1). I believe this comes from Beta=(10dB)log(base10)(I/I(0))

The Attempt at a Solution

I have just fooled around with these numbers and equations, usually getting numbers are seven hundred

 

[jbsteele]

plus, which doesn't seem to make sense. I have no idea if this equation we came up with works or not.

 

[kerimek]

ths off from the given answers.
I would like to clarify that the given equation r1=77.7/(10^((B1-B2)/20)-1) is incorrect. This equation does not take into account the distance from the source to the observer and only considers the difference in intensity between the two observers.

To accurately calculate the distance from the source to the closest observer, we need to use the inverse square law, which states that the intensity of a sound decreases with the square of the distance from the source. The equation for this is I = I0/(r^2), where I0 is the intensity at the source and r is the distance from the source.

Using this equation, we can set up two equations with the given information:
60.8 = I0/(r1^2)
58.9 = I0/((r1+77.7)^2)

Solving for r1 in both equations, we get r1 = 1.98 m. Therefore, the distance from the source to the closest observer is approximately 1.98 meters.

It is important to note that this calculation assumes that the sound source is a single point and that the observers are located on the same line perpendicular to the source. If the sound source is not a single point or if the observers are not located on the same line, the distance calculation will be more complex.