Height of a tower from two angles of elevation

Points \(A\) and \(B\) lie on level ground in a straight line with the foot \(F\) of a vertical tower of height \(h\) m, with \(B\) between \(A\) and \(F\) and \(AB=d\) m. The angles of elevation of the top of the tower from \(A\) and \(B\) are \(\alpha\) and \(\beta\) respectively, where \(\alpha<\beta\).

1. Show that \(h=\dfrac{d\tan\alpha\tan\beta}{\tan\beta-\tan\alpha}\).[3]
2. Given that \(d=20\), \(\alpha=30^\circ\) and \(\beta=45^\circ\), find the height of the tower.[2]
3. The angles in (b) were measured with an instrument held \(1.5\) m above the ground. State the assumption made in (b) and find a corrected height of the tower.[2]