Question

A machine shop has 120 equally noisy machines that together produce an intensity level of 92 dB.

If the intensity level must be reduced to 89 dB , how many machines must be turned off?

Answer 1

As you have read in your physics class -

The decibel (dB) scale is logarithmic, such that

dB = 10log(I/I0)

Where,

I is the intensity of the sound in W/m^2
I0 is the threshold for human hearing, or 10^(-12)W/m^2

so the difference between a sounds of 89 and 92 dB:

92dB - 89dB = 10log(I2/10) - 10 log(I1/10) where I2, I1 are the intensities of the two sounds

=> 3 db = 10 [log(I2/I0)-log(I1/I0)]

remember that log(a/b) = loga - log b so that

=> 3 db = 10[log(I2/I1)]

So, the ratio of intensities is

0.3=log(I2/I1) =>I2/I1 = 1.995

which means that the intensity of 89dB is 1/1.995 = 0.501 the intensity of 92 db

And this means you have 0.501*120machines = 50 machines to turn off.