Question

You have your headphones turned all the way up (100 dB).

Answer 1

Answer:

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