Calculate Distance between A & B: Sound Intensity Problem

[ubiquinone]

Hi I have a question here from a chapter on sound. I'm not sure on how to solve this, so I was wondering if someone here could please give me a hand. Thank You!

Question: A point source at A emits sound uniformly in all directions. At point B, the listener measures the sound intensity to be 50.0dB. At point C, the sound intensity level is 45.3dB. The distance \overline{BC} is 10.0m. Calculate the distance \overline{AB}.

Diagram:

A----------B
 \         |
   \       |
     \     |
       \   |
         \ |
           C

 

[Hootenanny]

Have you any thoughts yourself?

 

[ubiquinone]

This question is from an old physics exercise book which does not have any examples. I have tried solving it by reading up the concepts on waves and sound from a conceptual physics book. I'm hoping if someone here can show me how to solve this problem, so I can see and understand how a physicist or a good problem solver applies physics concepts and theory into problem solving. Again, any help with this would be greatly appreciated. Thanks!

 

[Hootenanny]

Perhaps this ; http://hyperphysics.phy-astr.gsu.edu/hbase/acoustic/invsqs.html#c1 would be good background reading.

 

[ubiquinone]

Hi Hootenanny, thank you for the reference! I think I got it now.

Let x be the distance of \overline{AB}
Calculate the intensities:
\displaystyle 50.0dB=10\log\left (\frac{I_1}{10^{-12}}\right )
\displaystyle\Leftrightarrow 10^5=\frac{I_1}{10^-12}
\displaystyle\Leftrightarrow 10^{-7}=I_1
\displaystyle 45.3dB=10\log\left (\frac{I_2}{10^{-12}}\right )
\displaystyle\Leftrightarrow 10^{4.53}=\frac{I_2}{10^-12}
\displaystyle\Leftrightarrow 10^{-7.47}=I_2
Since \displaystyle\frac{I_1}{I_2}=\frac{r_2^2}{r_1^2}\Rightarrow \frac{10^{-7}}{10^{-7.47}}=\frac{\sqrt{x^2+10^2}^2}{x^2}
\displaystyle\Leftrightarrow 10^{0.47}x^2=x^2+100
\displaysytle\Leftrightarrow x^2(-1+10^{0.47})=100
\displaystyle\Rightarrow x=\sqrt{\frac{100}{10^{0.47}-1}}\approx 7.16m