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.

The diagram shows a river,500m wide between the straight parallel banks where AD = 500m and DB = 2000. A man swims across the river from A to C at 0.5 m/s and then he runs along the banks from C to B at 0.8 m/s

(a)If he swims in the direction AC making an angle of Ɵ and reaches C in t seconds, show that t = 1000cosec Ɵ

(b)If T(in seconds) is the total time taken from A to B, express T in terms of Ɵ

(c)Find the minimum value of T in minutes and seconds, correct to the nearest second.

(a)

Using trigo ratio,

sin Ɵ = 500/AC

AC = 500/sin Ɵ

= 500cosec Ɵ

0.5 = 500cosec Ɵ/t

T = 1000cosec Ɵ

(b)

BC = 2000 – DC

BC = 2000 – 500/tan Ɵ

T_{BC }= BC/s = 2500 – 625/tan Ɵ

Total time, T = t_{AC} + t_{BC}

= 1000cosec Ɵ + 2500 – 625/tan Ɵ

(c)