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You have three tuning forks, A, B, and C. Fork B has a frequency of 489Hz ; when A and B are sounded together, a beat frequency of 3Hz is heard. When B and C are sounded together, the beat frequency is 5Hz . What are the possible frequencies of A? What are the possible frequencies of C? What beat frequencies are possible when A and C are sounded together?
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fb = 489 Hz

When A and B are sounded together,

fbeat = |fb - fa| = 3Hz

Thus the possible frequencies of fork A are:

fb - fa = 3Hz. Thi gives fa = 489 - 3 = 486 Hz

OR

fa - fb = 3Hz. This gives fa = 489 + 3 = 492 Hz

When B and C are sounded together,

fbeat = |fb - fc| = 5Hz

Thus the possible frequencies of fork C are:

fb - fc = 5Hz. This gives fc = 489 - 5 = 484 Hz

OR

fc - fb = 5Hz. This gives fa = 489 + 5 = 494 Hz

Now the possible combinations if A and C sounded together are:

1. fa = 486, fc = 484

Thus fbeat = 486 - 484 = 2 Hz

2. fa = 486, fc = 494

Thus fbeat = 494 - 486 = 8 Hz

3. fa = 492, fc = 484

Thus fbeat = 492 - 484 = 8 Hz

4. fa = 492, fc = 494

Thus fbeat = 494 - 492 = 2 Hz

Thus the possible beat frequencies when A and C are sounded together are 2 Hz and 8 Hz.
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