Question

You are trying to overhear a most interesting conversation, but from your distance of 10.0 m m , it sounds like only an average whisper of 20.0 dB d B . So you decide to move closer to give the conversation a sound level of 60.0 dB d B instead. How close should you come? The values of intensity at the points of interest are not given explicitly. Instead you are given the values of the intensity level at those points. Consturct a convenient expression for the intensity I that corresponds to an intensity level β with respect to a reference intensity I0. Express your answer in terms of some or all of the variables β and I0.

I don't think this question involves numbers but more as an expression with β and I0.

Answer 1

To convert a 20.0 dB whisper to 60.0 dB, understand that every 10 dB increase corresponds to a tenfold increase in intensity. Use the formula β (dB) = 10 log10 (I/I0), and consider that sound intensity decreases with the square of the distance from the source.

Step-by-step explanation:

To determine the distance you need to move to increase the sound level from 20.0 dB to 60.0 dB, it's first necessary to understand the relationship between sound intensity level (β) in decibels and sound intensity (I) in watts per meter squared. This relationship is given by the formula β (dB) = 10 log10 (I/I0), where I0 is the reference intensity of 10-12 W/m2, which corresponds to the threshold of human hearing.

An important point to keep in mind is that every increase of 10 dB represents a tenfold increase in intensity. Thus, an increase from 20.0 dB to 60.0 dB is an increase of 40 dB, which means the final intensity is 10,000 times greater than the initial intensity (as 40 dB corresponds to 104 times the intensity). Since sound intensity decreases with the square of the distance from the source, you can use the inverse square law to calculate how close you need to come. Formally, the expression for intensity is I = (β (dB) - 10 log10(I0)) / 10, solving for I in terms of β and I0.