Hi I am absolutely stumped on this question.
A loudspeaker is placed between two observers who are 110 m apart, along the line connecting them. If one observer records a sound level of 60.1 dB and the other records a sound level of 74.9 dB, how far is the speaker from each observer?
What I did for this is
I found the the Intensities of both the sound levels.
Then using the formula r1= SQRT(I2/I1)*R2
I would solve for r1 and then do r1+r2=113 to find r2.
I also don't know what value to assign for I2 and I1, i tried both ways but its wrong.
Please Help!
Thanks In Advance
duplicate post; already answered
To find the distance between the loudspeaker and each observer, you can use the inverse square law for sound intensity:
I₂/I₁ = (r₁/r₂)²
Where I₁ and I₂ are the intensities of the sound levels, and r₁ and r₂ are the distances between the loudspeaker and each observer, respectively.
In this case, you need to find the values of r₁ and r₂.
Here's how you can approach solving the problem step by step:
1. Convert the sound levels from decibels (dB) to intensities.
Remember that sound intensity is measured on a logarithmic scale, so you need to convert the sound levels to intensities before using the inverse square law equation. The conversion formula is:
I = 10^(L/10)
Where I is the intensity in watts per square meter (W/m²), and L is the sound level in decibels.
For the first observer:
I₁ = 10^(60.1/10)
I₁ ≈ 1.00 W/m²
For the second observer:
I₂ = 10^(74.9/10)
I₂ ≈ 31.62 W/m²
2. Plug the values of I₁, I₂, and the equation into the equation.
(r₁/r₂)² = I₂/I₁
(r₁/r₂)² = 31.62/1.00
(r₁/r₂)² = 31.62
3. Solve for the ratio of distances (r₁/r₂).
Take the square root of both sides of the equation:
r₁/r₂ = √31.62
r₁/r₂ ≈ 5.63
4. Solve for r₁ or r₂ using the ratio of distances.
Since you have two variables and one equation, you need a second equation to solve for r₁ and r₂ separately. The problem states that the distance between the observers is 110 m, so you can set up another equation:
r₁ + r₂ = 110
Now you have a system of equations:
r₁/r₂ = 5.63
r₁ + r₂ = 110
5. Solve the system of equations for r₁ and r₂.
To solve the system of equations, you can use various methods, such as substitution or elimination. Here's an example using substitution:
From the first equation, solve for r₁:
r₁ = 5.63r₂
Substitute this expression into the second equation:
5.63r₂ + r₂ = 110
6.63r₂ = 110
r₂ ≈ 16.59 m
Now substitute this value back into the first equation to find r₁:
r₁ = 5.63 * 16.59
r₁ ≈ 93.42 m
So, the loudspeaker is approximately 93.42 m from the first observer, and 16.59 m from the second observer.