Heights and Distances

-The angle of elevation of the top of a TV tower from three points $A, B$ and $C$ in a straight line through the foot of the tower are $\alpha, 2 \alpha$ and $3 \alpha$ respectively. If $A B=a$, then height of the tower…

From an aeroplane flying, vertically above a horizontal road, the angles of depression of two consecutive stones on the same side of the aeroplane are observed to be $30^{\circ}$ and $60^{\circ}$ respectively. The…

An aeroplane flying with uniform speed horizontally one $\mathrm{km}$ above the ground is observed at an elevation of $60^{\circ}$. After $10 \mathrm{~s}$ if the elevation is observed to be $30^{\circ}$, then the speed…

The angle of elevation of an object from a point $P$ on the level ground is $\alpha$. Moving $d$ metres on the ground towards the object, the angle of elevation is found to be $\beta$. Then the height (in metres) of the…

A vertical pole subtends an angle $\tan ^{-1}\left(\frac{1}{2}\right)$ at a point $P$ on the ground. If the angles substended by the upper half and the lower half of the pole at $P$ are respectively $\alpha$ and…

The sum of angles of elevation of the top of a tower from two points distant $a$ and $b$ from the base and in the same straight line with it is $90^{\circ}$. Then, the height of the tower is

From a point on the level ground, the angle of elevation of the top of a pole is $30^{\circ}$ on moving 20 metres nearer, the angle of elevation is $45^{\circ}$. Then, the height of the pole (in metres), is

From the top of a hill $h$ metres high the angles of depressions of the top and the bottom of a pillar are $\alpha$ and $\beta$ respectively. The height (in metres) of the pillar is

The angle of elevation of a stationary cloud from a point $2500 \mathrm{~m}$ above a lake is $15^{\circ}$ and from the same point the angle of depression of its reflection in the lake is $45^{\circ}$. The height (in…

A man \(2 \mathrm{~m}\) tall, walks at the rate of \(1 \frac{2}{3} \mathrm{~m} / \mathrm{s}\) towards a street light which is \(5 \frac{1}{3} \mathrm{~m}\) above the ground. The rate at which the length of his shadow is…

The elevation of an object on a hill is observed from a certain point in the horizontal plane through its base, to be $30^{\circ}$. After walking 120 metres towards it on level ground the elevation is found to be…

$P$ is a point on the segment joining the feet of two vertical poles of heights $a$ and $b$. The angles of elevation of the tops of the poles from $P$ are $45^{\circ}$ each. Then, the square of the distance between the…