Derive and calibrate a logistic population model

A town has population $3000$. The number infected after $t$ days satisfies $dI/dt=kI(3000-I)$, $k>0$.

(i) Given $I(0)=30$, show $I=3000/(1+99e^{-3000kt})$.6 marks

(ii) Given $I(7)=240$, find when $90\%$ of the population is infected.3 marks

(iii) State one contextual assumption needed for the model.1 mark