Question 2

Consider a water tank that is being filled with rainfall at the top and is drained through a small hole at

the bottom. The change in volume can be modelled by considering the amount of water entering and

leaving the tank (per unit of time) as follows

dV

dt

= fin - fout.

Here volume of water in the tank V is in litres, flow rate in fin is in litres per hour, flow rate out fout is

in litres per hour and time t is in hours. Initially, the tank contains 50L of water: V(0) = 50.

(a) (2 marks) Assume the rainfall throughout the day is getting heavier, such that, fin (t) = 10+t and

that the tank is losing 10% of its volume of water per hour, such that, fout (V) = 0.1V. Check by

direct substitution that the function

V(t) = 10(t+5e-0.1t)

satisfies the ODE and the initial condition.

(b) (4 marks) Now assume that the hole is slowly growing in size, so that the flow rate out increases

with time:

fout (V, t) = 0.1(1 + 0.1t)V.

Solve the ordinary differential equation for V(t) with this flow rate out, using either separation of

variables, or the integrating factor method. Make sure to also include the initial condition.

(c) (2 marks) use MATLAB to create a plot of both solutions from Q2a and Q2b (in the same figure).

You can either use the explicit solutions that come from solving the ODEs, or use ode45 to generate

numerical solutions to the ODEs. Make sure you have chosen an interval to plot over so that the

behaviour of the solutions can be clearly seen.