Question

The loudness, L, measured in decibels (Db), of a sound intensity, I, measured in watts per square meter, is defined as L = 10 log StartFraction I Over I 0 EndFraction, where I 0 = 10 Superscript negative 12 and is the least intense sound a human ear can hear. Brandon is trying to take a nap, and he can barely hear his neighbor mowing the lawn. The sound intensity level that Brandon can hear is 10-10. Ahmad, Brandon’s neighbor that lives across the street, is mowing the lawn, and the sound intensity level of the mower is 10-4. How does Brandon’s sound intensity level compare to Ahmad’s mower?
Brandon’s sound intensity is One-fourth the level of Ahmad’s mower.
Brandon’s sound intensity is One-fourth the level of Ahmad’s mower.
Brandon’s sound intensity is 20 times the level of Ahmad’s mower.
Brandon’s sound intensity is 80 times the level of Ahmad’s mower.

Answer 1

Interpret the given numbers as intensities: I_B = 10^-10 W/m^2 and I_A = 10^-4 W/m^2. Using L = 10 log(I/I0) with I0 = 10^-12:

L_B = 10 log(10^-10 / 10^-12) = 10 log(10^2) = 10·2 = 20 dB
L_A = 10 log(10^-4 / 10^-12) = 10 log(10^8) = 10·8 = 80 dB

So Brandon’s sound intensity level (20 dB) is one-fourth of Ahmad’s mower’s level (80 dB).