(i) The coordinates of \(P\) are \((t, 5t)\) since it lies on the line \(y = 5x\). The coordinates of \(Q\) are \((t, t(9 - t^2))\) since it lies on the curve \(y = x(9 - x^2)\). The length of \(PQ\) is the difference in the \(y\)-coordinates of \(Q\) and \(P\):

\(PQ = t(9 - t^2) - 5t = 9t - t^3 - 5t = 4t - t^3\)

(ii) To find the maximum value of \(PQ\), differentiate \(PQ = 4t - t^3\) with respect to \(t\):

\(\frac{d(PQ)}{dt} = 4 - 3t^2\)

Set the derivative to zero to find critical points:

\(4 - 3t^2 = 0\)

\(3t^2 = 4\)

\(t^2 = \frac{4}{3}\)

\(t = \frac{2}{\sqrt{3}}\)

Substitute \(t = \frac{2}{\sqrt{3}}\) back into \(PQ = 4t - t^3\) to find the maximum length:

\(PQ = 4\left(\frac{2}{\sqrt{3}}\right) - \left(\frac{2}{\sqrt{3}}\right)^3\)

\(PQ = \frac{8}{\sqrt{3}} - \frac{8}{3\sqrt{3}}\)

\(PQ = \frac{24}{3\sqrt{3}} - \frac{8}{3\sqrt{3}}\)

\(PQ = \frac{16}{3\sqrt{3}}\)

Alternatively, \(PQ = \frac{16\sqrt{3}}{9}\).