The rate of increase in the number of bacteria in a culture, \(N\), at time \(t\) hours is proportional to \(N\).
Write down a differential equation connecting \(N\) and \(t\).
Given that initially there are \(N _ { 0 }\) bacteria present in a culture,
Show that \(N = N _ { 0 } \mathrm { e } ^ { k t }\), where \(k\) is a positive constant.
Given also that the number of bacteria present doubles every six hours,
find the value of \(k\),
find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute.