When the angle of elevation of the sun increases from $30^{\circ}$ to $60^{\circ},$ the shadow of a tower decreases by $50\, m .$ Find the height of the tower in $m$.

From a point $h$ metres above the water level of a lake,the angle of elevation of the top of a palace is found to be $\alpha$ and the angle of depression of the image of the top of the palace observed in the lake is found to be $\beta$. Find the height of the palace.

There is a $100\, m$ high tower at some distance from a cliff. The angle of elevation of the top of the cliff from the base of the tower is $45^{\circ}$ and the angle of depression of the base of the cliff from the top of the tower is $30^{\circ}$. Find the height of the cliff and also the distance between the tower and the cliff.

The angle of depression of the base of a tower from the top of a cliff is $60^{\circ}$ and the angle of elevation of the top of the tower from the base of the cliff is $30^{\circ}$. If the height of the tower is $50 \, m$,find the height of the cliff in meters.

The height of a building is $11 \, m$ and the height of a pole is $8 \, m$. Watching from the midpoint of the line segment joining their bases,the angle of elevation of the top of the pole is $\alpha$ and that of the top of the building is $\beta$. Then,$\ldots \ldots \ldots$ holds true.

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 $m$).