Solve the following differential equations using the substitutions given.

(a)

(b)

A rectangular tank with its base horizontal is filled with water to a depth *h *at time *t* = 0. Water leaks out of the tank from a small hole in the base at a rate proportional to the square root of the depth of the water. If the depth of water is ½*h* at time* T,* find the further time, in terms of *T*, it will take before the tank is empty.

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.