9 The number of organisms in a population at time \(t\) is denoted by \(x\). Treating \(x\) as a continuous variable, the differential equation satisfied by \(x\) and \(t\) is $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x \mathrm { e } ^ { - t } } { k + \mathrm { e } ^ { - t } }$$ where \(k\) is a positive constant.

1. Given that \(x = 10\) when \(t = 0\), solve the differential equation, obtaining a relation between \(x , k\) and \(t\).
2. Given also that \(x = 20\) when \(t = 1\), show that \(k = 1 - \frac { 2 } { \mathrm { e } }\).
3. Show that the number of organisms never reaches 48, however large \(t\) becomes.