I am reposting this since I did so incorrectly the first time. (I apologize for posting it twice the first time as well...I accidently hit refresh) Here is the problem with the proper units:
You are trying to overhear a juicy conversation, but from your distance of 15.0m , it sounds like only an average whisper of 35.0 decibels . So you decide to move closer to give the conversation a sound level of 60.0 decibels instead. How close should you come? (in cm)
I Have no idea where to start with this problem....I don't even need an answer, but an equation to work with would be great. My college professor seemed to skip over decibels, but I'm having a hard time understanding it on my own. Thank you.
Well, the power going out of the mouth is the same, what is changing is the area that power is spread over at some distance. Notice that area in a sphere (assuming the sound goes equally in all directions) changes by distance^2
dblevel= 10 log (P/Po)=10log(radiuso/radius)^2=20log (original distance/new distance)
So you want a 25 db increase (35 to 60)
25= 20 log (15/distance)
10^(25/20)= 15/distance
distance= 15/10^1.25=84 cm
check my thinking.
the answer is C.
To solve this problem, we can use the equation for the sound intensity level, which is measured in decibels (dB):
β = 10 * log₁₀(I/I₀)
Where:
β is the sound intensity level in decibels (dB)
I is the sound intensity of the whisper in watts per square meter (W/m²)
I₀ is the threshold of hearing, which is approximately 1.0 x 10⁻¹² W/m²
Let's break down the problem and solve it step by step:
1. We are given the initial sound level of 35.0 decibels (β₁) at a distance of 15.0 meters.
2. We need to find the distance at which the sound level would be 60.0 decibels (β₂).
3. Using the equation for sound intensity level, we can set up two equations:
β₁ = 10 * log₁₀(I₁/I₀)
β₂ = 10 * log₁₀(I₂/I₀)
We can rearrange the equations to solve for I₁ and I₂:
I₁ = I₀ * 10^(β₁/10)
I₂ = I₀ * 10^(β₂/10)
4. Now we need to relate the two intensities, I₁ and I₂, to the distances from the sound source.
The relationship between sound intensity and distance is given by the inverse square law:
I₁/I₂ = (d₂/d₁)²
where d₁ is the initial distance (15.0 m), d₂ is the distance to be determined, and I₁ and I₂ are the respective sound intensities.
5. Substitute the expressions for I₁ and I₂ into the inverse square law equation:
(I₀ * 10^(β₁/10)) / (I₀ * 10^(β₂/10)) = (d₂/d₁)²
Simplify the equation by canceling out the common terms:
10^(β₁-β₂)/10 = (d₂/d₁)²
Take the square root of both sides:
sqrt(10^(β₁-β₂)/10) = d₂/d₁
Rearrange the equation to solve for d₂:
d₂ = d₁ * sqrt(10^(β₁-β₂)/10)
6. Convert the result to centimeters by multiplying by 100:
d₂_cm = d₂ * 100
Plug in the given values for β₁, β₂, and d₁ into the equation, and calculate d₂_cm. This will give you the required distance in centimeters.