The angle of elevation of building $Y$ from the bottom of building $X$ is $45^{\circ}$ and the angle of elevation of building $X$ from the bottom of building $Y$ is $65^{\circ}.$ Then:

$A$ $15\ m$ high eucalyptus tree standing erect on the ground breaks at the height of $5\ m$ from the ground. The broken part bends such that the top of the tree touches the ground. Find the angle made by the broken part of the tree with the ground. (in $^{\circ}$)

Two ships are anchored in the opposite directions from a lighthouse. The angle of depression of one ship from the top of the lighthouse is $60^{\circ}$ and that of the other ship from the same place is found to be $30^{\circ}$. If the ship with the angle of depression of $60^{\circ}$ is $150\,m$ away from the lighthouse,find the distance between these two ships (in $m$).

$A$ vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height $h$. At a point on the plane,the angles of elevation of the bottom and the top of the flag staff are $\alpha$ and $\beta$,respectively. Prove that the height of the tower is $\left(\frac{h \tan \alpha}{\tan \beta-\tan \alpha}\right)$.

$A$ person is watching from a point $A$ on the ground,which is $x \ m$ away from the base of a building. The height of the building is $y \ m$ and the angle of elevation of the top of the building is found to be $70^\circ$. Then:

As observed from the top of a tower $30 \, m$ high,the angle of depression of a stone on the ground is found to be $45^{\circ}$. Find the distance between the stone and the tower (in $m$).