IB Math HL Trigonometry Test 3 2014 P2 Q6

Points \(A,B,C\) lie in the same horizontal plane and \(AD\) is vertical. \(\angle ABC=45^\circ\), \(BC=50\text{ m}\). The angles of elevation from \(B\) and \(C\) to \(D\) are \(30^\circ\) and \(20^\circ\), respectively.

Using the cosine rule in triangle \(ABC\), or otherwise, find \(AD\).[6]