Radius, grazing area and angle for a tethered goat from an arc length, and a new fence found with bearings and the sine rule

A straight wall and a straight hedge meet at the point \(A\), and a lawn lies between them. A goat is tied to a peg \(P\) at the foot of the wall on a rope \(r\) metres long. The point \(D\) lies on the wall and the point \(E\) lies on the hedge, where \(PA=PD=PE=r\) and angle \(DPE=\theta\) radians, as shown in the diagram.

The arc \(DE\), centre \(P\), shown in the diagram has length \(30\) metres. The goat can graze the region bounded by the wall, the hedge and this arc.

1. Write \(r\) as an expression in \(\theta\).[1]
2. Show that the grazing area available to the goat is \(\dfrac{450}{\theta^2}(\theta+\sin\theta)\ \text{m}^2\).[4]
3. The goat can graze an area of \(520\ \text{m}^2\). Find \(\theta\).[2]
4. Hence, find the size of angle \(DAE\).[2]

To close off the lawn, a new straight fence \(BC\) is to be built, where \(B\) lies on the line of the wall and \(C\) lies on the line of the hedge, as shown in the second diagram. \(C\) is due east of \(B\), \(AB=700\) metres and the bearing of \(C\) from \(A\) is \(160^\circ\).

5. 1. Find the size of angle \(ACB\).
   2. Calculate the length of the new fence \(BC\).[5]