A middle C note (262 Hz) is sounded on a piano to help tune a violin string. When the string is sounded, nine beats are heard in 3.0 seconds. How much is the violin string off-tune?

To determine how much the violin string is off tune, we can calculate the difference between the middle C note played on the piano (262 Hz) and the frequency of the violin string. We are given that nine beats are heard in 3.0 s when the string is sounded. Since beats are created by the interference of two sound waves with slightly different frequencies, we can calculate the beat frequency and use it to find the frequency of the violin string.

First, we can use the relationship between beat frequency (f_beat), frequency of one of the waves (f1), and frequency of the other wave (f2):

f_beat = |f1 - f2|

In this case, f1 is the frequency of the middle C note on the piano (262 Hz) and f2 is the frequency of the violin string. We are given that f_beat is 9 beats in 3.0 s, so we can calculate the beat frequency:

f_beat = 9 beats / 3.0 s = 3 Hz

We know that the beat frequency is the absolute difference between the two frequencies, so we can set up the equation:

3 Hz = |262 Hz - f2|

Solving for f2, we can have two cases: f2 = 265 Hz or f2 = 259 Hz. Since the violin string is said to be off tune, we can assume that the frequency is higher than the piano's note. Therefore, the violin string is off tune by approximately 3 Hz higher than the middle C note on the piano.

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