Two boys are whispering in the library. The radiated sound power from one boy's mouth is 1.2 × 10^-⁹ W; and it spreads out uniformly in all directions. What is the minimum distance the boys must be away from the librarian so that she will not be able to hear them? The threshold of hearing for the librarian is 1.00 × 10^-¹² W/m². A)100 m B)35 m C)23 m D)16 m E)9.8 m

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Quizplus · Accepted Answer

The intensity of the sound wave at a distance r from the source is given by:$$I = \frac{P}{4\pi r^2}$$where P is the power of the sound source. The threshold of hearing is the minimum intensity of sound that can be heard by a person. In this case, the threshold of hearing is 1.00 × 10^-12 W/m^2. We can use this to find the minimum distance the boys must be away from the librarian so that she will not be able to hear them:$$1.00 	imes 10^{-12} W/m^2 = \frac{1.2 	imes 10^{-9} W}{4\pi r^2}$$$$r^2 = \frac{1.2 	imes 10^{-9} W}{4\pi (1.00 	imes 10^{-12} W/m^2)}$$$$r = \sqrt{\frac{1.2 	imes 10^{-9} W}{4\pi (1.00 	imes 10^{-12} W/m^2)}} = 9.8 m$$Therefore, the minimum distance the boys must be away from the librarian so that she will not be able to hear them is 9.8 m.

Two Boys Are Whispering in the Library