9 The population of a country at time \(t\) years is \(N\) millions. At any time, \(N\) is assumed to increase at a rate proportional to the product of \(N\) and \(( 1 - 0.01 N )\). When \(t = 0 , N = 20\) and \(\frac { \mathrm { d } N } { \mathrm {~d} t } = 0.32\).
- Treating \(N\) and \(t\) as continuous variables, show that they satisfy the differential equation
$$\frac { \mathrm { d } N } { \mathrm {~d} t } = 0.02 N ( 1 - 0.01 N )$$
- Solve the differential equation, obtaining an expression for \(t\) in terms of \(N\).
- Find the time at which the population will be double its value at \(t = 0\).