\includegraphics{figure_15}
Two tanks, A and B, each have a capacity of 800 litres.
At time \(t = 0\) both tanks are full of pure water.
When \(t > 0\), water flows in the following ways:
• Water with a salt concentration of \(\mu\) grams per litre flows into tank A at a constant rate
• Water flows from tank A to tank B at a rate of 16 litres per minute
• Water flows from tank B to tank A at a rate of \(r\) litres per minute
• Water flows out of tank B through a waste pipe
• The amount of water in each tank remains at 800 litres.
At time \(t\) minutes (\(t \geq 0\)) there are \(x\) grams of salt in tank A and \(y\) grams of salt in tank B.
This system is represented by the coupled differential equations
\begin{align}
\frac{dx}{dt} &= 36 - 0.02x + 0.005y \tag{1}
\frac{dy}{dt} &= 0.02x - 0.02y \tag{2}
\end{align}
- Find the value of \(r\).
[2 marks]
- Show that \(\mu = 3\)
[3 marks]
- Solve the coupled differential equations to find both \(x\) and \(y\) in terms of \(t\).
[9 marks]