A piano tuner compares a piano string with a 256 Hz tuning fork and hears 2 beats per second. What are the possible frequencies of the piano string?

1. Identify the knowns: f_B = 2 Hz, and one frequency, the tuning fork, is f2 = 256 Hz. 2. The beat frequency formula f_B = |f1 - f2| can be satisfied two ways: f_B = f1 - f2, or f_B = -(f1 - f2). 3. Solving for f1 in the first case: f1 = f_B + f2 = 2 Hz + 256 Hz = 258 Hz. 4. Solving for f1 in the second case: f1 = -f_B + f2 = -2 Hz + 256 Hz = 254 Hz. 5. So the piano string could be at either 258 Hz or 254 Hz, matching option A. 6. Option C, 512 Hz or 128 Hz, looks like an octave doubling or halving error rather than a correct use of the 2 Hz beat frequency. 7. Options B and D both use frequency gaps that are too large to come from a 2 Hz beat frequency against this tuning fork. _Source: OpenStax Physics (CC BY 4.0), Ch 14 "Sound", section Resonance and Beats_