You are trying to overhear a juicy conversation, but from your distance of 17.0 m, it sounds like only an average whisper of 35.0 dB. So you decide to move closer to give the conversation a sound level of 80.0 dB instead.
How close should you come?
d= ?? cm
halving the distance adds six db. You can use the inverse square law to show that.
halving the distance = 41db
halving again=47db
halving again= 53 db
or (1/2)^n= 6n= 80-35
(1/2)^n=45
2^-n=45
log2 of each side
n=log2 45= 6.7
distance= 17*2^-6.7=.162m or 16.2cm
check my thinking
it didn't end up being correct, but i'm not sure where it goes wrong...
To determine how close you should come, we need to understand the relationship between distance and sound intensity.
Sound intensity decreases with increasing distance from the source. The relationship between sound intensity (I) and distance (d) is described by the inverse square law:
I1 / I2 = (d2 / d1)^2
Where:
I1 is the initial sound intensity (at the initial distance d1)
I2 is the final sound intensity (at the final distance d2)
In this case, we have initial sound intensity (I1) at a distance (d1) of 17.0 m and final sound intensity (I2) of 80.0 dB.
We need to convert the decibel scale to sound intensity using the equation:
I2 = I0 * 10^(dB/10)
Where:
I2 is the sound intensity in Watts per square meter (W/m^2)
I0 is the reference sound intensity (about 10^-12 W/m^2)
dB is the decibel value
First, let's convert the final sound intensity of 80.0 dB to Watts per square meter.
I2 = I0 * 10^(80.0/10)
I2 = 10^(-12) * 10^8
I2 = 10^(-12+8)
I2 = 10^-4 W/m^2
Now, we can use the inverse square law to find the final distance (d2) from the source:
(d2^2 / d1^2) = I1 / I2
(d2^2 / (17.0^2)) = 35.0 / 10^-4
d2^2 = (17.0^2) * (35.0 / 10^-4)
d2^2 = (289) * (35.0 / 10^-4)
d2^2 = (289) * (35.0) * 10^4
d2^2 = 289 * 35 * 10^4
d2^2 = 101150000
Taking the square root of both sides, we get:
d2 = sqrt(101150000)
d2 ≈ 10058.49 cm
Therefore, you should move closer to approximately 10058.49 cm or 100.58 m (rounded to two decimal places) to hear the conversation at a sound level of 80.0 dB.