Question

A family ice show is held at an enclosed arena. The skaters perform to music with level 80.0dB. This level is too loud for your baby, who yells at 75.0 dB. (a) What total sound intensity engulfs you? (b) what is the combined sound level ?

Answer 1

Resist the temptation to say 155 dB. The decibel scale is logarithmic, so addition does not work that way. We must figure out the intensity of each sound, add the intensities, and then translate back to a level. We have
`beta=10log((I)/(10^(-12)(W)/(m^2)))`
`I=[10^((beta)/(10))]10^(-12)(W)/(m^2)`
a. for your baby,
`I_b=(10^((75.0)/(10)))(10^(-12)(W)(m^2)=sqrt(10)xx10^_5(W)/(m^2)`
For the music, `I_m=(10^((80.0)/(10)))(10^(-12)(W)/(m^2))=10.0xx10^-5(W)/(m^2)`
The combined intesity is
`I_(total)=I_m+I_b`
`=10.0xx10^-5(W)/(m^2)+3.16xx10^-5(W)/(m^2)`
`=13.2xx10^-5(W)/(m^2)`
b. The combined sound level is then
`beta_(total)=10log((I_(total))/(10^(-12)(W)/(m^2)))`
`=10log((1.32xx10^(-4)(W)/(m^2))/(10^(-12)(W)/(m^2)))=81.2dB`
The result is only a little louder than 80 dB.