5. The number of bacteria present in a culture at time \(t\) hours is modelled by the continuous variable \(N\) and the relationship
$$N = 2000 \mathrm { e } ^ { k t } ,$$
where \(k\) is a constant.
Given that when \(t = 3 , N = 18000\), find
- the value of \(k\) to 3 significant figures,
- how long it takes for the number of bacteria present to double, giving your answer to the nearest minute,
- the rate at which the number of bacteria is increasing when \(t = 3\).