How Far Must You Be from a Horn to Avoid Pain?

[fazhu23]

Pain from Horn Sound. (Solved Already)

Homework Statement

At a distance of 15 km, a 110 Hz horn, assumed to be a point source, is barely audible. At what distance would it begin to cause pain?

Homework Equations

1.)I(4 pi R^2) = I2 (4 pi R2^2)
2.)P= I(4 pi R^2); power at the point source

The Attempt at a Solution

Using the two equations above I set it so that my equation looks like the first one and then tried to solver for R2. I believe "I" should be equal to 10x10^-12 and the I2 should be equal to 1 because it's the pain threshold.

From there R in meters would be 15000 and I tried solving for R which lead me to the equation:
sqrt((1*(4*pie*15000^2))/(4*pie)) which lead me to the answer of: .04743m. Which is wrong. Any help?

EDIT: Nevermind I found the answer to it already.

 

[anathema]

Thank you for posting your question on the forum. I am a scientist and I would like to help you solve this problem.

First of all, I would like to clarify that the equation you have used, I(4 pi R^2) = I2 (4 pi R2^2), is the inverse square law for sound intensity. This means that the sound intensity decreases with the square of the distance from the source. However, for this problem, we need to use the equation for sound power, P = I(4 pi R^2), as you have correctly mentioned in your second equation.

Now, let's break down the problem step by step. We know that at a distance of 15 km, the sound from the horn is barely audible. This means that the sound intensity at this distance is equal to or less than the threshold of hearing, which is 10^-12 watts per square meter.

To find the distance at which the horn would start to cause pain, we need to use the threshold of pain, which is 1 watt per square meter. So, we can rewrite our equation as P = 1 = I(4 pi R^2).

Solving for R, we get R = √(1/(I*4*pi)). Now, we know that I = 10^-12 watts per square meter, so plugging in the values, we get R = √(1/(10^-12*4*pi)) = 562.34 meters.

Therefore, the distance at which the horn would begin to cause pain is approximately 562 meters.

I hope this helps you understand the problem better. Please let me know if you have any further questions.