The carrying capacity to the hectare for a population \(P\) of a flock of sheep is calculated to be 100 sheep to the hectare. The logistic differential equation for \(P\) is given by

\[\frac{dP}{dt} = 0.01 P(100-P) \text{ at time } t \text{ years.}\]

(i) Show that \(P = \frac{100}{1+Ke^{-t}}\) is a solution to the logistic differential equation.

(ii) If initially \(P=10\), find \(K\) and hence the population when \(t = 5\).