Answer:
5.59 times per second.
Step-by-step explanation:
Direct variation is in the form:
[tex]y=kx[/tex]
Where k is the constant of variation.
Inverse variation is in the form:
[tex]\displaystyle y=\frac{k}{x}[/tex]
In the given problem, the frequency of a vibrating guitar string varies inversely as its length. In other words, using f for frequency and l for length:
[tex]\displaystyle f=\frac{k}{\ell}[/tex]
We can solve for the constant of variation. We know that the frequency f is 4.3 when the length is 0.65 meters long. Thus:
[tex]\displaystyle 4.3=\frac{k}{0.65}[/tex]
Solve for k:
[tex]k=4.3(0.65)=2.795[/tex]
So, our equation becomes:
[tex]\displaystyle f=\frac{2.795}{\ell}[/tex]
Then when the length is 0.5 meters, the frequency will be:
[tex]\displaystyle f=\frac{2.795}{.5}=5.59\text{ times per second.}[/tex]
