9709 P33 - Jun 2015 - Q7
2291
The number of micro-organisms in a population at time \(t\) is denoted by \(M\). At any time, the variation in \(M\) is assumed to satisfy the differential equation
\[ \frac{dM}{dt} = k\sqrt{M}\cos(0.02t), \]
where \(k\) is a constant and \(M\) is taken to be a continuous variable. It is given that when \(t = 0\), \(M = 100\).
- Solve the differential equation, obtaining a relation between \(M\), \(k\), and \(t\).
- Given also that \(M = 196\) when \(t = 50\), find the value of \(k\).
- Obtain an expression for \(M\) in terms of \(t\) and find the least possible number of micro-organisms.
