Learning Goal:

To understand how to convert between different sound intensity scales and how the decibel intensity of a sound changes with distance.

The decibel scale is logarithmic in intensity:

$=10$logII$0$$=10$logII$0.$

In this formula,$$I$0$I$0$is a reference intensity, which, for sound waves, is taken to be$1012$W$/$m$21012$W$/$m$2.$ This constant must be used to convert a particular physical intensity into a sound intensity level measured in decibels.

Once we know the sound intensity level $($in decibels$)$ at a certain reference distance from a sound source, the$1/$r$21/$r$2$decrease of intensity with distance can be accounted for by$$subtracting$$the decibel value appropriate to the ratio of the new distance to the reference distance.

In this problem you will use the decibel scale to analyze a small firecracker that emits $1200$WW$$of peak power. To avoid confusion, intensities denoted by$$II$$are in units of watts per meter squared; intensities denoted by$$$$are in units of decibels.

It takes a sound intensity of about $160$ dB to rupture the human eardrum. How close must the firecracker described in the introduction be to the ear to rupture the eardrum?

Express the distance$$DruptureDrupture$,$ in meters, to one decimal place.