You are 10 m away from a sound source of
intensity comparable to the background
noise in a very quiet room (20 dB). If you
want the intensity in dB to double, you must
move to a distance ...
A) 100 m
B) 20 m
C) 7 m
D) 5 m
E) 1 m
the answer is 1 m, but I'm not sure how to approach this question. I've tried using these 2 formulas, but I'm not really getting anywhere:
--> I α 1/r^2
--> I(dB) = 10log(I/Io)
Please help!
20 db = 10 log (I/Io)
so
log (I/Io) = 2
so
I/Io = 10^2
now do 40 db
40 db = 10 log (I/Io)
log I/Io = 4
I/Io = 10^4
the ratio of intensities = 10^4/10^2 =100
so
r^2/(10^2) = 1/100
r^2 = 1
r = 1
If you had said "Physics" instead if "please help" I would have seen this earlier.
To solve this question, we need to understand how sound intensity changes with distance and how to calculate decibels.
First, let's recall the relationship between sound intensity and distance. The sound intensity (I) is inversely proportional to the square of the distance (r) from the source. Mathematically, this can be expressed as I α 1/r^2, where α denotes proportionality.
Now, let's move on to understanding decibels (dB). Decibels are a logarithmic unit used to quantitatively express sound intensity. The formula to calculate decibels is given by: I(dB) = 10log(I/Io), where I is the sound intensity being measured and Io is the reference intensity set at 10^(-12) watts/m^2.
Now, let's work through the problem. The question states that you are initially 10 m away from a sound source with an intensity comparable to the background noise in a very quiet room, which is given as 20 dB.
To double the sound intensity in dB, we need to find the new distance (let's call it r') from the sound source.
To solve for r', we can equate the two decibel values and solve for r':
10log(I/Io) = 10log(I'/Io)
Since we are only interested in doubling the intensity, we can simplify the equation to:
10log(I) = 10log(I')
Cancelling out the common factor of 10, we are left with:
log(I) = log(I')
Taking the anti-log (inverse logarithm) of both sides, the equation becomes:
I = I'
This means that the initial intensity (I) must be equal to the final intensity (I').
In our case, the initial intensity (I) is given by the background noise in a very quiet room, which is 10^(-12) watts/m^2. So, to double this intensity, the new intensity (I') must also be 10^(-12) watts/m^2.
Using the formula I α 1/r^2, we can set up the equation:
10^(-12) α 1/(r')^2
Rearranging the equation, we get:
(r')^2 = 1/(10^(-12))
Taking the square root of both sides:
r' = 1 m
Therefore, to double the intensity from the background noise level, you must move to a distance of 1 m from the sound source.
Hence, the correct answer is E) 1 m.