(i) Separate variables and integrate:

\(\int 2x^2 \, dx = \int \frac{k - x^3}{t} \, dt\)

Integrate to obtain:

\(\ln t = -\frac{2}{3} \ln(k - x^3) + \frac{2}{3} \ln(k - 1)\)

Use initial conditions \(t = 1, x = 1\) and \(t = 4, x = 2\) to solve for \(k\):

\(\ln 4 = -\frac{2}{3} \ln(k - 8) + \frac{2}{3} \ln(k - 1)\)

Solve for \(k\) to obtain \(k = 9\).

Substitute \(k = 9\) to find \(x\):

\(x = (9 - 8t^{\frac{3}{2}})^{\frac{1}{3}}\)

(ii) As \(t \to \infty\), \(x \to 9^{\frac{1}{3}}\).