(i) The velocity of the particle is given by \(v = 0.03t^2\). To find the displacement \(x(t)\), we integrate the velocity function:

\(x(t) = \int v \, dt = \int 0.03t^2 \, dt = 0.01t^3 + C\)

Given that when \(t = 5\), \(x = 2.5\), we substitute to find \(C\):

\(2.5 = 0.01(5)^3 + C\)

\(2.5 = 1.25 + C\)

\(C = 1.25\)

Thus, the expression for displacement is \(x = 0.01t^3 + 1.25\).

(ii) We need to find the velocity when the displacement \(x = 11.25\):

\(0.01t^3 + 1.25 = 11.25\)

\(0.01t^3 = 10\)

\(t^3 = 1000\)

\(t = 10\)

Substitute \(t = 10\) into the velocity equation:

\(v = 0.03(10)^2 = 3 \text{ m/s}\)