Chapter 12, Problem 70GP

To determine

To Find: The acoustic power output of the speaker.

Answer to Problem 70GP

The acoustic power output of the speaker is $P=57.22\text{W}$ .

Explanation of Solution

Given:

Sound level $r=\text{ 12m}$ ,

Sound intensity $I =\text{1}0\text{5 dB}$

Formula used:

The power output of a speaker is expressed as:

  $P=I.4\pi r^2\text{ W}$

Where $r$ is the sound level.

I = Intensity of sound

The intensity level of sound that can be produced from a loudspeaker is expressed as.

  $\beta =10{\log }_{10}\left(\frac{I}{I_0}\right)$

Here, $I$ is sound intensity,

  $I_0$ is the hearing intensity.

Calculation:

Power output (P) of the speaker can be calculated by using the intensity of sound that can be produced from a loudspeaker.

  $\beta =10{\log }_{10}\left(\frac{I}{I_0}\right)$

I = Intensity of sound = $I={10}^{10.5}{I}_0$

  $I_0=1\times {10}^{-12}$ W/m²$\beta =105\text{ dB}$$\beta =10\text{ dB}$

Now, substitute all the values in the Equation of Power, P

  $P=I.4\pi r^2\text{ W}$

  $P={10}^{10.5}\times 1\times {10}^{-12}(4\pi \times {12}^2)$

  $P=57.22\text{ W}$

Conclusion:

The acoustic power output is 57.22 W, of the speaker, assuming it radiates equally in all directions.