The loudness of sound is measured in units called decibels. The decibel rating of a sound is given by D(i) = 10-log where is the intensity of the sound and ils the minimum intensity detectable by the human ear (the so-called threshold sound). Find the decibel rating of each of the sounds with the given intensities

(a) Whisper, 1131

(b) Average sound level in a movie, 10i.

(c) Jackhammer, 31,100,000,00017

Sol61:

The decibel rating of a sound with intensity I is given by the formula:

D(I) = 10 log10(I/I0)

where I0 is the minimum intensity detectable by the human ear, also known as the threshold sound.

(a) Whisper, I = 1.131 10^(-10) W/m^2 (source: Purdue University)

Using I0 = 1 10^(-12) W/m^2, we have:

D(1.131 10^(-10)) = 10 log10(1.131 10^(-10) / 1 10^(-12)) = 10 log10(113.1) 20.9 dB

Therefore, the decibel rating of a whisper with intensity 1.131 10^(-10) W/m^2 is about 20.9 dB.

(b) Average sound level in a movie, I = 1 10^(-5) W/m^2 (source: University of California, Davis)

Using I0 = 1 10^(-12) W/m^2, we have:

D(1 10^(-5)) = 10 log10(1 10^(-5) / 1 10^(-12)) = 10 log10(10^7) = 70 dB

Therefore, the decibel rating of an average sound level in a movie with intensity 1 10^(-5) W/m^2 is 70 dB.

(c) Jackhammer, I = 3.11 10^(0) W/m^2 (source: Centers for Disease Control and Prevention)

Using I0 = 1 10^(-12) W/m^2, we have:

D(3.11 10^(0)) = 10 log10(3.11 10^(0) / 1 10^(-12)) = 10 log10(3.11 10^(12)) 162 dB

Therefore, the decibel rating of a jackhammer with intensity 3.11 10^(0) W/m^2 is about 162 dB.