Answer
a. 122 dB, 115 dB.
b. No.
Work Step by Step
a. First find the intensity of the sound, which is power per unit area. Assume that the sound spreads out as a spherical wave from each loudspeaker.
$$I_{expensive}=\frac{220\;W}{4\pi(3.5\;m)^2}=1.429\;W/m^2$$
$$\beta_{expensive}=10log\frac{I_{expensive}}{I_o}=10log\frac{1.429\;W/m^2}{1.0\times10^{-12}W/m^2}\approx 122\;dB$$
$$I_{cheap}=\frac{45\;W}{4\pi(3.5\;m)^2}=0.292\;W/m^2$$
$$\beta_{cheap}=10log\frac{I_{cheap}}{I_o}=10log\frac{0.292\;W/m^2}{1.0\times10^{-12}W/m^2}\approx 115\;dB$$
b. The textbook claims that 2 sound levels must differ by 10 dB for the louder sound to be judged twice as loud. The sounds differ by only about 7 dB, so the expensive one doesn’t sound twice as loud as the cheap one.
