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------------------------------------------------ The length of a violin string, [tex]l[/tex], varies inversely as the frequency of its vibrations, [tex]f[/tex]. When the length increases, the frequency decreases, and when the length decreases, the frequency increases. A violin string 18 inches long vibrates at a frequency of 350 cycles per second.

Find the frequency of a 15-inch violin string.

[tex]f = \square[/tex]

To solve this problem, let's recall the concept of inverse variation. If two quantities are inversely proportional, their product is constant. In this case, the length of the violin string [tex]$ l $[/tex] varies inversely with the frequency [tex]$ f $[/tex]. This relationship can be expressed as:

[tex]\[ l \times f = k \][/tex]

where [tex]$ k $[/tex] is the constant.

We can find the constant using the given information for the 18-inch string:

1. Given:
- Length of the string, [tex]$ l_1 = 18 $[/tex] inches
- Frequency of the string, [tex]$ f_1 = 350 $[/tex] cycles per second

2. Calculate the constant [tex]$ k $[/tex]:
[tex]\[
k = l_1 \times f_1 = 18 \times 350
\][/tex]

3. Now, find the frequency for the new 15-inch string. Let the new frequency be [tex]$ f_2 $[/tex], and the new length [tex]$ l_2 = 15 $[/tex] inches.

4. Set up the equation using the constant:
[tex]\[
l_2 \times f_2 = k
\][/tex]

5. Substitute the known values:
[tex]\[
15 \times f_2 = 18 \times 350
\][/tex]

6. Solve for [tex]$ f_2 $[/tex]:
[tex]\[
f_2 = \frac{18 \times 350}{15}
\][/tex]

7. Calculate the frequency:
[tex]\[
f_2 = 420
\][/tex]

So, the frequency of the 15-inch violin string is 420 cycles per second.