Question

What would be the intensity level of a 39-decibel sound after being amplified a million times?

Answer 1

Given

Intensity in Decibel 39

Increase in intensity million times

Known values

Threshold intensity of hearing I_o = 1 x 10^-12 W/m²

Solution

Intensity in Decibel I_dB = 10 log{(Intensity in w/m²)/Threshold Intensity}

39 dB = 10 log(I/I_o)

39 = 10log(I/I_o)

3.9 = log(I/I_o)

10^3.9 = I/I_o

I = I_o x 10^3.9

After amplifying million times the new intensity

I’ = million times I

I’ = 1 x 10⁶ x I

I’ = 1 x 10⁶ x I_o x 10^3.9

I’ = I_o x 10^9.9

I’ in decibels

I’ dB = 10 log(I’/I_o)

I’ dB = 10 log(I_o x 10^9.9/I_o)

I’ dB = 10 log(10^9.9)

I’dB = 10 x 9.9

I’ dB = 99 Decibels

After amplifying one million times the intensity level of 39 dB sound would be 99 dB