The total number of infected trees, $x$, is modelled by

$$x=A\times B^t$$

where $A$ and $B$ are constants and $t$ is the number of days after the farmer first noticed the infection.

(i) Find the total number of infected trees 6 days after the farmer first noticed the infection based on this model.

(ii) Explain why this model is not realistic in the long term for the orchard.

A refined model assumes the rate of increase of the number of infected trees is given by

$$\frac{dx}{dt}=\frac{x(1200-x)}{4800}$$

(i) Show that

$$\int \left(\frac{C}{x}+\frac{D}{1200-x}\right)dx=\int dt$$

where $C$ and $D$ are positive integers to be found.

(ii) Hence, find $t$ in terms of $x$.

(iii) Find the number of days it takes from when the infection is first noticed until half of the trees are infected.