A researcher is investigating the spread of a certain disease in a town with a population of 3000 people. The researcher suggests that *I*, the number of people infected by the disease at time *t *days satisfies the differential equation , where *k* is a positive constant.

**(i) **Given that *I *= 30 when *t *= 0, show that .

**(ii) **It is further observed that *I *= 240 when *t *= 7 find the time it takes for 90% of the population to be infected by the disease.

**(iii) **State, in the context of this question, one assumption needed to model the spread of the disease in the town by the given differential equation.

(iii) Possible Answer:

Assume that the total population of the town is 3000 during the spread of the disease.

Or: Assume that a person infected by the disease will remain infected by the disease.

Or: Assume that everyone in the town has an equal chance of being infected by the disease.