Watching from the top of a $150\, m$ 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 $45^{\circ}$ respectively. Find the distance between the houses in meters.

$A$ ladder leans on a wall to reach a height of $3 \, m$ on the wall. If the ladder makes an angle of $30^{\circ}$ with the ground,the length of the ladder is ........... $m$.

$A$ 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.

The angle of elevation of a cloud from a point $h$ metres above the surface of a lake is $\theta$ and the angle of depression of its reflection in the lake is $\phi$. Prove that the height of the cloud above the lake is $h\left(\frac{\tan \phi+\tan \theta}{\tan \phi-\tan \theta}\right).$

$A$ $4 \, m$ long ladder leans on a wall such that its lower end remains $2 \, m$ away from the base of the wall. Then,the ladder makes an angle of measure $\ldots \ldots$ with the ground. (in $^\circ$)

Find the angle of elevation of the sun when the shadow of a pole $h$ meters high is $\sqrt{3} h$ meters long. (in $^{\circ}$)