Given that \(x\) varies directly as the square root of \(y\), we can express this relationship as:

\[x = k \cdot \sqrt{y}\]

where \(k\) is the constant of variation.

We are given that when \(x = 81\) and \(y = 9\), we can substitute these values into the equation to find \(k\):

\[81 = k \cdot \sqrt{9}\]

\[81 = k \cdot 3\]

\[k = \frac{81}{3}\]

\[k = 27\]

Now that we have the constant of variation \(k = 27\), we can use it to find \(x\) when \(y = 1\frac{7}{9}\):

\[x = 27 \cdot \sqrt{1\frac{7}{9}}\]

\[x = 27 \cdot \sqrt{\frac{16}{9}}\]

\[x = 27 \cdot \frac{4}{3}\]

\[x = 36\]

So, when \(y = 1\frac{7}{9}\), \(x = 36\).