Question

1. the length of a violin string varies inversely as the frequency of its vibrations. a violin string 14 inches long vibrates at a frequency of 450 cycles per second. find the frequency of a 12 - inch violin string.

Explanation:

Step1: Define inverse variation

Let \( l \) be the length of the string and \( f \) be the frequency. For inverse variation, \( l = \frac{k}{f} \) (or \( lf = k \), where \( k \) is the constant of variation).

Step2: Find the constant \( k \)

We know that when \( l = 14 \) inches, \( f = 450 \) cycles per second. Substitute into \( lf = k \): \( k = 14\times450 = 6300 \).

Step3: Find the frequency for \( l = 12 \) inches

Now, use \( lf = k \) with \( l = 12 \) and \( k = 6300 \). So, \( 12f = 6300 \). Solve for \( f \): \( f=\frac{6300}{12}= 525 \).

Answer:

The frequency of a 12 - inch violin string is 525 cycles per second.