Crane-jib extension and roof clearance

Core stretch

A crane stands in a warehouse 22 m high. Its pivot \(A\) is 9 m above ground, and its jib \(AB=16\) m is inclined \(30^\circ\) above horizontal \(AC\). A vertical cable from \(B\) reaches ground at \(D\).

1. Find \(BD\).[2]
2. The jib is extended by 4 m and raised by a further \(15^\circ\). Determine whether its tip will hit the roof. Give the new tip height and roof clearance to 2 decimal places.[2]