Question

Speakers loudness. Measured in decibel (dB) is inversely proportional to the square of the distance d of the listener from the speaker.If the loudness is 20 dB when the listener is 6 feet away from the speaker.What is loudness when the listener is 3 ft away from the speaker

Answer 1

Let Speaker Loudness be represented by L

Let the distance of the listener from the speaker be represented by d

`L\text{ }\propto\text{ }(1)/(d^2)`

`L\text{ = }(k)/(d^2)`

we have to get the constant of proportionaity, k first

We were told that, When L = 20dB, d = 6 feet

`\begin{gathered} 20\text{ = }(k)/(6^2) \\ 20\text{ =}(k)/(36) \\ \\ k\text{ = 36 x 20} \\ k\text{ = 720} \end{gathered}`

Then we can find L, when d = 3 feet

`\begin{gathered} L\text{ =}(k)/(d^2) \\ k\text{ = 720 , d =3} \\ L=(720)/(3^2) \\ L=\text{ }(720)/(9) \\ \\ L\text{ =80 dB} \end{gathered}`