8 The variables \(x\) and \(t\) satisfy the differential equation
$$t \frac { \mathrm {~d} x } { \mathrm {~d} t } = \frac { k - x ^ { 3 } } { 2 x ^ { 2 } }$$
for \(t > 0\), where \(k\) is a constant. When \(t = 1 , x = 1\) and when \(t = 4 , x = 2\).
- Solve the differential equation, finding the value of \(k\) and obtaining an expression for \(x\) in terms of \(t\).
- State what happens to the value of \(x\) as \(t\) becomes large.