As the number of people at a party increases you must raise

As the number of people at a party increases, you must raise your voice for a listener to hear you against the background noise of the other partygoers. However, once you reach the level of yelling, the only way you can be heard is if you move closer to your listener, into the listener\'s \"personal space\". Model the situation by replacing you with an isotropic point source of fixed power p and replacing your listener with a point that absorbs part of your sound waves. These points are initially separated by ? = 1.4m. If the background noise increases by Delta beta = 5.4 dB, the sound level at your listener must also increase. what separation r_t is then required?

Solution

L_f - L_i = 5.4 dB

Sound intensity level in dB is L = 10 * log( I / I)

10 * log( I_f / I) - 1₀ * log( Ii / I) = 5.4

log( I_f / I_i ) = 0.54

The sound intensity I = P / (4 pi r²)

log( r_i² / r_f² ) = 0.54

2 * log( r_i / r_f ) = 0.54

r_i/r_f = 10^0.27

r_f = r_I / 10^0.27 = 1.40 m / 10^0.27

= 0.752 m