A water dispenser contains 10 litres of water initially. Water is poured into the dispenser at a constant rate of \(4\cos\theta\) litres per minute, for some positive real value of \(\theta\). At the same time, water is dispensed out from the dispenser at a constant rate of \(6\sin\theta\) litres per minute.
Explain why the value of \(\theta\) cannot fall in the range between \(\dfrac\pi2\) to \(2\pi\).[1]
Find the value of \(\theta\), where \(6<\theta<8\), for which the volume of water in the dispenser remains constant.[3]
If, instead, the water in the dispenser is emptied after 4 minutes, show that \(3\sin\theta-2\cos\theta=1.25\).[2]
By writing \(3\sin\theta-2\cos\theta\) in the form \(R\sin(\theta-\alpha)\), where \(R>0\) and \(\alpha\) is acute, find the volume of water poured into the dispenser during the 4 minutes.[5]