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Two musicians are comparing their trombones. The first produces a tone that is known to be 438 Hz. When the two trombones play together they produce 6 beats every 2 seconds. Which statement is true about the second trombone?

It is producing a 444-Hz sound, and could be producing no other sound frequency.
It is producing either a 436-Hz sound or a 440-Hz sound.
It is producing either a 435-Hz sound or a 441-Hz sound.
It is producing either a 432-Hz sound or a 444-Hz sound.
It is producing a 441-Hz sound, and could be producing no other sound frequency.

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skyluke89

Answer:

It is producing either a 435-Hz sound or a 441-Hz sound.

Explanation:

Beats are produced when two waves (like the sound waves produced by the two trombones) having slightly difference frequency produce costructive interference. When this occurs, the resultant sound wave (the beats) has a frequency equal to the absolute difference between the frequencies of the individual waves:

[tex]f_B = |f_1 - f_2|[/tex] (1)

where [tex]f_B[/tex] is called beat frequency.

In this problem, we have:

[tex]f_1 = 438 Hz[/tex] is the frequency of the first trombone

[tex]f_B = \frac{6}{2 s}=3 Hz[/tex] is the beat frequency

Therefore, by solving (1) for [tex]f_2[/tex], we find two possible frequencies for the second trombone:

[tex]f_2 = f_1 - f_B = 438 Hz - 3 Hz = 435 Hz\\f_2 = f_1 + f_B = 438 Hz + 3 Hz = 441 Hz[/tex]

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