Answer
$\approx 21.33$ inches.
Work Step by Step
$y$ varies $\bf{inversely}$ with $x,$ or $y$ is inversely proportional to $x,$
if there is a nonzero constant $k$ such that $\displaystyle \ \ y=\frac{k}{x}$
---
Here, y= rate of vibration,
x = length of the string,
and $y=\displaystyle \frac{k}{x}$
Given that, for x=48, y=256, we calculate k:
$256=\displaystyle \frac{k}{48}\qquad/\times 48$
$k=12,288$
So, $y=\displaystyle \frac{12,288}{x}$.
When y=576, we find x from:
$576=\displaystyle \frac{12,288}{x}\qquad/\times\frac{x}{576}$
$x=\displaystyle \frac{12,288}{576}\approx 21.33$ inches.