You have your headphones turned all the way up (100 dB). a. If you want them to sound half as loud, to what decibel level must the music be set?

b. If you want them to sound 1/4 as loud, to what decibel level must the music be set?

a. 90 dB

b. 80 dB

Explanation:

The given parameter of the sound is as follows;

The level to which the headphones are tuned = All the way up

The intensity of the sound of the headphones when tuned all the way up = 100 dB

a. The relationship between level change in loudness, ΔL, and the ratio of loudness, 'x', is presented as follows;

ΔL = 10·log₂(x) ≈ 33.22·log(x)

Therefore, for a sound to be half as loud, we have x = 1/2, therefore;

ΔL = 10·log₂(1/2) = 10·log₂(2⁻¹) = -10 ≈ 33.22·log(1/2)

The change in the intensity level for a sound half as loud, ΔL = -10 dB

Given that the sound was initially at 100 dB, the new level for a sound half as loud = 100 dB + ΔL = 100 dB - 10 dB = 90 dB

The decibel level the music must be set for a sound half as loud = 90 dB

b. In order for the sound to be 1/4 as loud, we have;

ΔL = 10·log₂(1/4) = 10·log₂(2⁻²) = -20

Therefore, the change in the level of the sound intensity level, ΔL by -20 dB will given a sound that is 1/4 as loud

The initial intensity level = 100 dB

The intensity level for a sound 1/4 as loud = 100 dB - 20 dB = 80 dB