Techniques of Differentiation - Derivatives of Trigonometric Functions

The diagram shows the vertical cross-section \(PQRS\) of an open trough. The lengths of \(PQ\), \(QR\) and \(RS\) are \(30\) cm, \(40\) cm and \(30\) cm, respectively. The trough rests with \(QR\) as the base on horizontal ground and both \(PQ\) and \(RS\) are inclined at an acute angle \(\theta\) radians to the ground.

1. Show that the area, \(A\) \(\text{cm}^2\), of the cross-section \(PQRS\) is given by \(A = 1200 \sin \theta + 450 \sin 2\theta\).
2. Given that \(\theta\) can vary, find the value of \(\theta\) for which the volume of the trough is maximised.