In a certain chemical reaction, the amount, \(x\) grams, of a substance present is decreasing. The rate of decrease of \(x\) is proportional to the product of \(x\) and the time, \(t\) seconds, since the start of the reaction. Thus \(x\) and \(t\) satisfy the differential equation

\[ \frac{dx}{dt} = -kxt, \]

where \(k\) is a positive constant. At the start of the reaction, when \(t = 0\), \(x = 100\).

(i) Solve this differential equation, obtaining a relation between \(x\), \(k\), and \(t\).

(ii) Twenty seconds after the start of the reaction, the amount of substance present is 90 grams. Find the time after the start of the reaction at which the amount of substance present is 50 grams.