(i) Separate variables and integrate:

\(\int \frac{1}{\sqrt{M}} \, dM = \int k \cos(0.02t) \, dt\)

Integrating both sides gives:

\(2\sqrt{M} = 50k \sin(0.02t) + C\)

Using initial condition \(M = 100\) when \(t = 0\), we find \(C = 20\).

Thus, \(2\sqrt{M} = 50k \sin(0.02t) + 20\).

(ii) Given \(M = 196\) when \(t = 50\), substitute to find \(k\):

\(2\sqrt{196} = 50k \sin(1) + 20\)

\(28 = 50k \sin(1) + 20\)

\(8 = 50k \sin(1)\)

\(k = \frac{8}{50 \sin(1)} \approx 0.190\)

(iii) Express \(M\) in terms of \(t\):

\(2\sqrt{M} = 50 \times 0.190 \sin(0.02t) + 20\)

\(\sqrt{M} = 4.75 \sin(0.02t) + 10\)

\(M = (4.75 \sin(0.02t) + 10)^2\)

The least possible number of micro-organisms occurs when \(\sin(0.02t) = -1\):

\(M = (4.75(-1) + 10)^2 = (5.25)^2 = 27.5625\)

Thus, the least possible number is approximately 28 or 27.5 or 27.6.