Physics, 10.12.2019 02:31 lilay30

At a distance of 14.0 m from a point source, the intensity level is measured to be 80 db. at what distance from the source will the intensity level be 40 db

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Answer from: chevysilverado2377

1400m

Explanation:

For spherical waves we can use the following relationship between distance and intensity:

$\frac{I_{1}}{I_{2}}=\frac{r_{2}^2}{r_{1}^2}$

Where $I_{1}$ and $I_{2}$ are the first and second intensity, in W/m^2 . And $r_{1}$ is the first distance: $r_{1}=14m$ , and $r_{2}$ the distace we want to find.

Clearing the previous equation or $r_{2}$

$r_{2}^2=\frac{I_{1}r_{1}^2}{I_{2}} \\r_{2}=\sqrt{\frac{I_{1}r_{1}^2}{I_{2}}} \\r_{2}=r_{1}\sqrt{\frac{I_{1}}{I_{2}}}$ .

This is what we will be using to find the answer, but fist we must convert the quantities 80dB and 40dB to W/m^2

We will call the quantities in dB $\beta_{1}$ and $\beta _{2}$ :

$\beta_{1}=80dB$

$\beta_{2}=40dB$

We will use the following to find the corresponding intensities $I_{1}$ and $I_{2}$ :

$\beta_{1}=10log\frac{I_{1}}{I_{o}}$

$\beta_{2}=10log\frac{I_{2}}{I_{o}}$

for both of these: $I_{0}=10^{-12}W/m^2$ , and it is the minimum intensity that a human being perceives.

Thus, for $\beta_{1}=80dB$ we have:

$80dB=10log\frac{I_{1}}{I_{o}}$

$8dB=log\frac{I_{1}}{I_{o}}$

And to eliminate the logarithm, we use its inverse operation, exponentiation.

$10^8=10^{log\frac{I_{1}}{I_{0}} }$

$10^8=\frac{I_{1}}{I_{0}}$

$I_{0}(10^8)=I_{1}$

replacing $I_{0}=10^{-12}W/m^2$ :

$10^{-12}W/m^2(10^8)=I_{1}$

$1x10^{-4}W/m^2=I_{1}$

and similarly for $I_{2}$

$40dB=10log\frac{I_{2}}{I_{o}}$

$4dB=log\frac{I_{2}}{I_{o}}$

$10^4=10^{log\frac{I_{2}}{I_{0}} }$

$10^4=\frac{I_{2}}{I_{0}}$

$I_{0}(10^4)=I_{2}$

replacing $I_{0}=10^{-12}W/m^2$ :

$10^{-12}W/m^2(10^4)=I_{2}$

$1x10^{-8}W/m^2=I_{2}$

Now we substitute in the equation we had found for $r_{2}$

$r_{2}=r_{1}\sqrt{\frac{I_{1}}{I_{2}}}\\r_{2}=(14m)\sqrt{\frac{1x10^{-4}W/m^2}{1x10^{-8}W/m^2}}\\r_{2}=(14m)\sqrt{10000}\\ r_{2}=(14m)(100)\\r_{2}=1400m$

Answer from: Quest

heat the gas and measure the pressure

Answer from: Quest

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At a distance of 14.0 m from a point source, the intensity level is measured to be 80 db. at what di...