As observed from the top of a $510 \ m$ high tower,the angles of depression of two houses situated in the east and west of the tower are found to be $30^{\circ}$ and $60^{\circ}$ respectively. Find the distance between the two houses (in $m$).

$A$ window of a house is $h$ meters above the ground. From the window,the angles of elevation and depression of the top and the bottom of another house situated on the opposite side of the lane are found to be $\alpha$ and $\beta$,respectively. Prove that the height of the other house is $h(1 + \tan \alpha \cot \beta)$ meters.

$A$ ladder is leaning against a wall such that its upper end touches the wall at a height of $3 \, m$ and the ladder is inclined at an angle of $30^{\circ}$ with the ground. Find the length of the ladder in $m$.

$A$ spherical balloon of radius $r$ subtends an angle $\theta$ at the eye of an observer. If the angle of elevation of its centre is $\phi$,find the height of the centre of the balloon.

$A$ pole stands erect on the ground. $A$ wire tied to the top of the pole is affixed at a point on the ground. If the length of the wire is $7 \, m$ and the wire makes an angle of $30^{\circ}$ with the ground,find the height of the pole (in $m$).

The angle of depression of a car parked on the road as observed from the top of a building is $60^{\circ}$. The angle of depression of that car from a window $14.6 \, m$ below the top is $45^{\circ}$. Find the distance between the car and the building (in $m$).