In the afternoon, the decibel level of a busy freeway is 80dB with 100 cars passing a given point every minute.
Late at night, the traffic flow is only 5 cars per minute. What is the late-night decibel level???
The decibel level `dB` of a sound shows on a logarithmic scale the intensity `I` of that sound.
`dB = 10*log(I/I_0)`
where `I_0=10^-12 W/m^2` is the sound intensity reference.
Also, the intensity of a sound (`I`) is defined as the power of the sound over the unit area.
Since power is, in turn defined as total work done over a given time,
it means that sound power (and in turn the intensity of sound) is a medium value over a given time interval for sound produced by one or multiple sources.
Assuming every car is making the same noise, the intensity level of the sound will be proportional to the total number of cars passing in a given interval of time (because is a medium value over time).
For 100 cars/minute in the afternoon the intensity of sound is
`I_1 = I_0*10^(80/10) =10^-12*10^8 =10^-4 W/m^2`
For 5 cars/minute late-night the intensity level is
`I_2 =I_1*5/100 =5/100*10^-4 =5*10^-6 W/m^2`
Therefore the decibel level of the sound late night is
`dB = 10*log(I_2/I_0) =10*log((5*10^-6)/10^-12) = 66.99 =70`
The late-night decibel level is 70 dB.